The scale factor used is 2.
What is triangle?It is one of the simplest polygon shapes and is commonly used in mathematics and geometry. The sum of the internal angles of a triangle is always 180 degrees.
Define vertices of triangle?The vertices of a triangle are the three points in a two-dimensional (2D) or three-dimensional (3D) space that define the corners or corners of the triangle. In a 2D plane, the vertices are typically denoted as A, B, and C, and in a 3D space, they can be represented as (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) where (x, y, z) are the coordinates of each vertex along the x, y, and z axes respectively. The vertices of a triangle are connected by three line segments, known as edges, to form the sides of the triangle. The combination of the three vertices and the edges connecting them determines the shape and size of the triangle.
To find the scale factor used to dilate triangle ABC to A'B'C', we can compare the corresponding side lengths of the two triangles.
The distance between A(-3, -3) and B(3, 3) is √((3-(-3))^2 + (3-(-3))^2) = 6√2.
The distance between A'(-6, -6) and B'(6, 6) is √((6-(-6))^2 + (6-(-6))^2) = 12√2.
So the scale factor used to dilate triangle ABC to A'B'C' is:
scale factor = length of corresponding side in A'B'C' / length of corresponding side in ABC
= (12√2) / (6√2)
= 2
Therefore, the scale factor used is 2.
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The point (-5,. 7) is located on the terminal arm of ZA in standard position. A) Determine the primary trigonometric ratios for ZA If applicable, make Sure yoU rationalize the denominator: b) Determine the primary trigonometric ratios for _B with the Same sine as ZA; but different signs for the other two primary trigonometric ratios If applicable, make sure you rationalize the denominator: c) Use a calculator to determine the measures of ZA and _B, to the nearest degree:
(a)We can use these values to calculate the primary trigonometric ratios:
sin(ZA) = o/h ≈ 0.139
cos(ZA) = a/h ≈ -0.998
tan(ZA) = o/a ≈ -0.14
(b) The same sine as ZA but different signs for the other two primary trigonometric ratios can be found by reflecting point (-5, 0.7) across the x-axis.
(c)We use inverse trigonometric functions on primary ratios ZA ≈ 7 degrees, B ≈ -7 degrees.
(a)How to calculate primary trigonometric ratios?To determine the primary trigonometric ratios for ZA, we first need to find the values of the adjacent, opposite, and hypotenuse sides of the right triangle that contains point (-5, 0.7) as one of its vertices. We can use the Pythagorean theorem to find the hypotenuse:
h = sqrt((-5)² + 0.7²) ≈ 5.02
The adjacent side is negative since the point is to the left of the origin, so:
a = -5
The opposite side is positive since the point is above the x-axis, so:
o = 0.7
Now we can use these values to calculate the primary trigonometric ratios:
sin(ZA) = o/h ≈ 0.139
cos(ZA) = a/h ≈ -0.998
tan(ZA) = o/a ≈ -0.14
(b) How trigonometric ratios can be found by reflecting point?To find a point B with the same sine as ZA but different signs for the other two primary trigonometric ratios, we can reflect point (-5, 0.7) across the x-axis. This gives us point (-5, -0.7), which has the same sine but opposite sign for the cosine and tangent:
sin(B) = sin(ZA) ≈ 0.139
cos(B) = -cos(ZA) ≈ 0.998
tan(B) = -tan(ZA) ≈ -0.14
(c) How to determine measures of nearest degree?To find the measures of ZA and B to the nearest degree, we can use inverse trigonometric functions on their primary ratios. Using a calculator, we get:
ZA ≈ 7 degrees
B ≈ -7 degrees (Note: this is equivalent to 353 degrees since angles are periodic).
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If the sides of a rectangle are in the ratio 3:4 and the length of the diagonal is 10 cm, find the length of the sides
Answer:
if the diagonal is 10 then the sides are 3*2 and 4*2 which is 6 and 8 respectively because the diagonal makes it a right angled triangle whereby the the 3,4,5 line steps in, so if the diagonal(hypotenuse) is 10 the 10/5 is 2 then you multiply both 3 and 4 by 2 and that gives you the length of two sides
6. (2.5 pts) at the beginning of week 5, they broke up. jack wanted to run off to the city with
diane, but diane said he was crazy. unfortunately, their relationship ended. both were
angry with each other. suppose we could somehow quantify and measure anger. let's
call the units "anger units". on the day of the break-up, jack had 100 anger units. every
week he lost 5% of his anger. recall that the growth factor needs to be the amount that
"stays on" jack (not the 5% that "comes off" jack). for example, after 1 week, he had 95
anger units. after 2 weeks he had 90.25 anger units, and so on. write an equation that
models jack's anger (let that be )) after t weeks.
We'll model Jack's anger in anger units after t weeks using an exponential decay equation, as he loses 5% of his anger every week.
To write an equation that models Jack's anger (let that be A(t)) after t weeks, we need to follow these steps:
1. Identify the initial amount of anger units (A0): Jack had 100 anger units at the beginning (t=0).
2. Determine the growth factor (1 - decay rate): Since Jack loses 5% of his anger every week, the growth factor is 1 - 0.05 = 0.95.
3. Set up the exponential decay equation: A(t) = A0 * (growth factor)^t.
By following these steps, the equation modeling Jack's anger after t weeks is:
A(t) = 100 * (0.95)^t
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The ratio of boys to girls in mrs. Cunninghams class is 2 to 3, there are 18 girls in the class. What is the total number of students in mrs. Cunninghams class
The total number of students in Mrs. Cunningham's class is 30.
From the question we know that the ratio of boys to girls in Mrs. Cunningham's class is 2 to 3 so we can write
no.of boys: no.of girls = 2:3
The total number of girls in the class is given as 18 so with this we can find out the number of boys in the class that is :
no.of boys= (2/3)*no.of girls in class
now after substituting the values in the equation, we get
no. of boys = (2/3) * 18
no.of boys = 12.
So, now we know the number of boys in the class that is 12 and the number of girls in the class is 18.
We can calculate the total number of students in the class which is equal to
= no.of boys + no.of girls.
= 12+18
=30
Therefore, the total number of students in Mrs. Cunningham's class is 30.
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A major corporation is building a 4,325 acre complex of homes, offices, stores, schools, and churches in the rural community of Glen Cove. As a result of this development, the planners have estimated that Glen Clove's population (in thousands) t years from now will be given by the following function.
P(t) = (45t^2 + 125t + 200)/t^2 + 6t + 40 (a) What is the current population (in number of people) of Glen Cove?
(b) What will be the population (in number of people) in the long run?
(a) To find the current population of Glen Cove, we need to substitute t = 0 in the given function.
P(0) = (45(0)^2 + 125(0) + 200)/(0)^2 + 6(0) + 40
P(0) = 200/40
P(0) = 5
Therefore, the current population of Glen Cove is 5,000 people (since the function is in thousands).
(b) To find the population in the long run, we need to take the limit of the function as t approaches infinity.
lim P(t) as t → ∞ = lim (45t^2 + 125t + 200)/(t^2 + 6t + 40) as t → ∞
Using L'Hopital's rule, we can find the limit of the numerator and denominator separately by taking the derivative of each.
lim P(t) as t → ∞ = lim (90t + 125)/(2t + 6) as t → ∞
Now, we can just plug in infinity for t to get the population in the long run.
lim P(t) as t → ∞ = (90∞ + 125)/(2∞ + 6)
lim P(t) as t → ∞ = ∞/∞ (since the numerator and denominator both go to infinity)
We can use L'Hopital's rule again to find the limit.
lim P(t) as t → ∞ = lim 90/2 as t → ∞
lim P(t) as t → ∞ = 45
Therefore, the population in the long run will be 45,000 people (since the function is in thousands).
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Create a bucket by rotating around the y axis the curve y = 4 ln(x - 4) from y = 0 to y = 3. If this bucket contains a liquid with density 860 kg/m filled to a height of 2 meters, find the work required to pump the liquid out of this bucket (over the top edge). Use 9.8 m/s2 for gravity. Work = Preview Joules License Points possible: 1 This is attempt 1 of 3.
The work required to pump the liquid out of the bucket is approximately 2.482 x 10⁷ Joules.
How to find the work requiredTo create a bucket by rotating around the y-axis, we will use the formula for volume of revolution:
V = π ∫[a,b] (f(y))² dy where f(y) is the function being rotated, and a and b are the limits of integration.
In this case, the limits of integration are y = 0 and y = 3, and the function being rotated is y = 4 ln(x - 4), or x = e⁽y/⁴⁾ + 4.
So, we have:
V = π ∫[0,3] ((e⁽y/⁴⁾ + 4))² dy
V = π ∫[0,3] (e⁽y/²⁾ + 8e⁽y/⁴⁾+ 16) dy
V = π (2e⁽³/²⁾ + 32e⁽³/⁴⁾ + 48)
Now, to find the work required to pump the liquid out of the bucket, we need to use the formula:
W = ∫[h1,h2] ρgV(y) dy
where h1 is the height of the liquid (2 meters in this case), h2 is the height of the top edge of the bucket, ρ is the density of the liquid (860 kg/m^3), g is the acceleration due to gravity (9.8 m/s²), and V(y) is the volume of the liquid at height y.
To find V(y), we need to first find the radius of the bucket at height y.
The radius is given by: r(y) = e⁽y/⁴⁾+ 4
So, the volume of the liquid at height y is:
V(y) = π(r(y))² (h2 - y)
Plugging in the values, we have:
W = ∫[0,2] 860×9.8×π((e⁽y/⁴ + 4)²)×(2-y) dy
W = 2.482 x 10⁷J
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I just need the answer to question 3
Answer:46.15% or rounded 46% of pulling a card higher than 8.
Step-by-step explantwenty. Well there are 6 cards higher than 8, which include 9, 10, jack, queen, king, and ace. There is 4 diffrent suites so do 6×4=24. Then do 24/52=0.4615
Answer: 46.15%
Which expression represents the surface area of the prism?
Choose 1 answer:
Choose 1 answer:
(Choice A)
2
⋅
6
+
12
+
8
+
8
2⋅6+12+8+82, dot, 6, plus, 12, plus, 8, plus, 8
A
2
⋅
6
+
12
+
8
+
8
2⋅6+12+8+82, dot, 6, plus, 12, plus, 8, plus, 8
(Choice B)
2
⋅
3
+
3
⋅
8
2⋅3+3⋅82, dot, 3, plus, 3, dot, 8
B
2
⋅
3
+
3
⋅
8
2⋅3+3⋅82, dot, 3, plus, 3, dot, 8
(Choice C)
3
+
3
+
12
+
8
+
8
3+3+12+8+83, plus, 3, plus, 12, plus, 8, plus, 8
C
3
+
3
+
12
+
8
+
8
3+3+12+8+83, plus, 3, plus, 12, plus, 8, plus, 8
(Choice D)
12
+
12
+
12
+
3
+
3
12+12+12+3+312, plus, 12, plus, 12, plus, 3, plus, 3
D
12
+
12
+
12
+
3
+
3
12+12+12+3+3
Options A and C are ruled out because they don't even represent legitimate expressions for a prism's surface area. [tex]12 + 12 + 12 + 3 + 3.[/tex]Thus, option D is correct.
What is the surface area of the prism?The expression that represents the surface area of the prism depends on the dimensions of the prism. However, we can use the formula for the surface area of a rectangular prism, which is:
Surface Area [tex]= 2lw + 2lh + 2wh[/tex]
where l is the length, w is the width, and h is the height of the prism.
Looking at the answer choices:
A)[tex]2.6 + 12 + 8 + 8 = 28.6[/tex]
B)[tex]2.3 + 3.8 = 6.1[/tex]
C)[tex]3 + 3 + 12 + 8 + 8 = 34[/tex]
D)[tex]12 + 12 + 12 + 3 + 3 = 42[/tex]
We can eliminate options A and C because they are not even valid expressions for the surface area of a prism.
Option B is a valid expression for the surface area, but it is not simplified.
Option D is also a valid expression for the surface area, and it is simplified.
Therefore, the answer is (D) [tex]12 + 12 + 12 + 3 + 3.[/tex]
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1 pts How much bubble wrap is needed to cover a cylindrical vase that is 16 inches tall with a diameter of 6 inches?
415 square inches of bubble wrap to cover the cylindrical vase that is 16 inches tall with a diameter of 6 inches.
To calculate how much bubble wrap is needed to cover the cylindrical vase, you will need to find the circumference and height of the vase.
First, calculate the circumference of the vase using the diameter of 6 inches:
Circumference = π x diameter
Circumference = 3.14 x 6
Circumference = 18.84 inches
Next, calculate the height of the vase which is given as 16 inches.
To find the surface area of the vase, you will need to multiply the circumference by the height and add the area of the circular bases. The formula for the surface area of a cylinder is:
Surface area = 2πr² + 2πrh
where r is the radius and h is the height.
Since the vase has circular bases, we can find the area of each base by using the formula:
Area of circle = πr²
Now, let's find the radius of the vase:
[tex]Radius = \frac{diameter}{2}[/tex]
[tex]Radius = \frac{6}{2}[/tex]
Radius = 3 inches
So, the area of each base is:
Area of base = π x (radius)²
Area of base = π x 3²
Area of base = 28.27 square inches
The total area of the two bases is 2 x 28.27 = 56.54 square inches.
Now, let's find the surface area of the cylinder:
Surface area = 2πr² + 2πrh
Surface area = 2 x π x 3² + 2 x π x 3 x 16
Surface area = 113.1 + 301.44
Surface area = 414.54 square inches
Therefore, you would need approximately 415 square inches of bubble wrap to cover the cylindrical vase that is 16 inches tall with a diameter of 6 inches.
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A right rectangular prism has length 10 in. And width 8 in. The surface area of the prism is 376 in2. What equation can be used to find the height in inches?
The equation to find the height is 376 = 160 + 20h + 16h.
The height of the right rectangular prism is 6 inches.
We have,
Let's denote the height of the right rectangular prism as "h" inches.
The formula for the surface area of a right rectangular prism is:
Surface Area = 2lw + 2lh + 2wh
Given that the length (l) is 10 inches and the width (w) is 8 inches, and the surface area is 376 square inches, we can substitute these values into the formula:
376 = 2(10)(8) + 2(10)(h) + 2(8)(h)
Simplifying this equation:
376 = 160 + 20h + 16h
Combine like terms:
376 = 160 + 36h
Rearranging the equation to isolate "h":
36h = 376 - 160
36h = 216
Finally, divide both sides of the equation by 36 to solve for "h":
h = 216/36
h = 6
Therefore,
The height of the right rectangular prism is 6 inches.
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1 point
6. The radius of the circular garden pond is 1. 75 feet. If a landscaper wants
to place a decorative fence around the circumference of the pond, about
how many feet of fencing will be needed? *
O 1. 099 feet
O 10. 99 feet
109. 9 feet
O 10. 99 square feet
O 1,099 square feet
The landscaper will need 10.99 feet of fencing to place around the circumference of the pond. Option B is the correct answer.
We need to find how many feet of the fence is needed to decorate the fence around the circumference of the pond. we can determine it by finding the circumference of a pound or circle. The circumference of a circle is calculated using the formula,
C = 2πr
Where:
C = the circumference
r = radius
Given data:
π = 3.14
r = 1. 75 feet.
Substuting the value of the radius in the formula we get
C = 2πr
C = 2π(1.75)
= 10.99
Therefore, the landscaper will need 10.99 feet of fencing to place around the circumference of the pond.
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The figure shows the graphs of the functions y=f(x) and y=g(x). If g(x)=kf(x), what is the value of k? Enter your answer in the box given.
The value of k is -2
Let a line passes through the point [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. Thus the equation of line can be given as,
[tex](y -y_1)=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]......Eq.(1)
We have the information from the graph:
The graph of f(x) and g(x) are given in the problem.
The equation given is,
g(x) = k × f(x)
We have to find the value of k and also find the equation of f(x) and g(x).
The line y = f(x) lies on the points, (2,1) and (0,-3). Thus the equation of this line is,
Plug all the values in eq.(1)
[tex](y -(-3))=\frac{-3-1}{0-2}(x-0)[/tex]
[tex]y+3=\frac{-4}{-2}x[/tex]
y + 3 = 2x
y = 2x -3
So, it can be written as:
f(x) = 2x -3
The line y = f(x) lies on the points, (0,6) and (2,-2). Thus the equation of this line is,
[tex](y -6)=\frac{-2-6}{2-0}(x-0)[/tex]
[tex](y-6)=\frac{-8}{2}x[/tex]
(y- 6) = -4x
y = -4x + 6
It can be written as:
g(x) = -4x + 6
The equation given in the problem is:
g(x) = k × f(x)
Put all the values in above given equation:
-4x + 6 = k(2x - 3)
-2(2x - 3) = k × (2x - 3)
Compare the value of k :
k = -2
Hence, The value of k = -2
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For complete question, to see the attachment:
Help me please I don’t know what to do
Answer:
179.3
Step-by-step explanation:
Rectangle:
L x W
10 x 14 = 140
Semicircle:
(π · r²) / 2
D = 10, r = 10 ÷ 2 = 5
(3.14 · 5²) / 2 = 39.25
Area of figure = 140 + 39.25 = 179.25 = 179.3 (rounding to tenth)
You are working as a financial planner. a couple has asked you to put together an investment plan for the education of their daughter. she is a bright seven-year-old (her birthday is today), and everyone hopes she will go to university after high school in 10 years, on her 17th birthday. you estimate that today the cost of a year of university is $17,500, including the cost of tuition, books, accommodation, food, and clothing. you forecast that the annual inflation rate will be 5. 6%. you may assume that these costs are incurred at the start of each university year. a typical university program lasts 4 years. the effective annual interest rate is 6. 75% and is nominal. a. suppose the couple invests money on her birthday, starting today and ending one year before she starts university. how much must they invest each year to have money to send their daughter to university? (do not round intermediate calculations. round your answer to 2 decimal places. )
investment per year $
b. if the couple waits 1 year, until their daughter’s 8th birthday, how much more do they need to invest annually? (do not round intermediate calculations. round your answer to 2 decimal places. )
additional yearly payments $
The couple needs to invest $9,060.52 per year to have enough money to send their daughter to university. The couple needs to invest an additional $1,322.18 per year if they wait one year to start saving for their daughter's university education.
a. The amount of money the couple needs to invest each year can be calculated using the present value of annuity formula. The future value of the university cost after 10 years can be calculated by compounding the current cost for 10 years at an annual inflation rate of 5.6%.
Then, the present value of this future cost can be found by discounting it back to the present using the effective annual interest rate of 6.75%. Finally, this present value can be divided by the present value of an annuity factor for 9 years (one year before the university starts) at an effective annual interest rate of 6.75%.
Using these calculations, the couple needs to invest $9,060.52 per year to have enough money to send their daughter to university.
b. If the couple waits for one year, they will have nine years to save for their daughter's university education. This means they will have one less year to invest, so they will need to invest more each year to have enough money for their daughter's university education.
The additional amount they need to invest can be found by subtracting the present value of an annuity of $9,060.52 for 9 years from the present value of an annuity of $9,060.52 for 8 years.
Using these calculations, the couple needs to invest an additional $1,322.18 per year if they wait one year to start saving for their daughter's university education.
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Imagine that the price per gallon of gas with a $7 car wash is $3. 19 and the price without the car wash is $3. 39. When is it worth it to buy the car wash? When is it worth it if the car wash costs $2?
if we need to purchase more than 10 gallons of gas, it is worth it to buy the car wash that costs $2.
When is it worth buying a $7 car wash with gas priced at $3.19 per gallon instead of buying gas without a car wash priced at $3.39 per gallon?
With a car wash, the price per gallon is $3.19, which is $0.20 less than the price without a car wash ($3.39).
To determine whether it is worth it to buy the car wash, we need to calculate the cost savings per gallon by purchasing the car wash.
Cost savings per gallon = Price without car wash – Price with car wash
Cost savings per gallon = $3.39 – $3.19 = $0.20
So, if the car wash costs less than $0.20 per gallon, it is worth it to purchase it.
If the car wash costs $2, we need to determine how many gallons of gas we need to purchase in order for the cost savings to be greater than $2.
Let's assume we purchase x gallons of gas. The cost savings for purchasing the car wash will be:
Cost savings = x gallons × $0.20 per gallon = $0.20x
We want to find the value of x that makes the cost savings greater than $2:
$0.20x > $2
x > $2 ÷ $0.20
x > 10
Therefore, if we need to purchase more than 10 gallons of gas, it is worth it to buy the car wash that costs $2.
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A recipe to make 4 pancakes calls for 6 teaspoon of flour. Tracy wants to make 10 pancakes using thks recipe. What equation will she needs to use to find out how many tablespoons of flour to use?
Thus, equation that Tracy needs to use to obtain the number of tablespoons of flour to use in making 10 pancakes.
Explain about the unitary method:The unitary method is a method for determining the value of one unit from the values of several units or the other way around.
The unitary approach is a strategy for problem-solving that involves first determining the value of one unit, then multiplying that value to determine the required value.
Given data:
4 pancakes ---> 6 teaspoon of flour.
For 1 pancake, divide above expression with 4 on both side.
1 pancakes ---> 6/4 teaspoon of flour.
Now, for 10 pancake, multiply above expression with 10 on both side.
10 pancakes ---> 10* 6/4 teaspoon of flour.
Thus, equation that Tracy needs to use to obtain the number of tablespoons of flour to use in making 10 pancakes.
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Shen will rent a car for the weekend. He can choose one of two plans. The first plan has an initial fee of $40 and costs an additional $0.30 per mile driven. The second plan has no initial fee but costs $0.80 per mile driven.
If Shen drives fewer than 80 miles, the first plan will be cheaper. If he drives more than 80 miles, the second plan will be cheaper.
Let's denote the number of miles driven by "m".
Under the first plan, Shen will pay an initial fee of $40, and then an additional $0.30 for each mile driven. So the total cost, C1, can be expressed as:
C1 = 0.3m + 40
Under the second plan, Shen will not have to pay an initial fee, but he will be charged $0.80 for each mile driven. So the total cost, C2, can be expressed as:
C2 = 0.8m
To determine which plan is cheaper for a given number of miles driven, we can set the two expressions for cost equal to each other and solve for "m":
0.3m + 40 = 0.8m
Subtracting 0.3m from both sides, we get:
40 = 0.5m
Dividing both sides by 0.5, we get:
m = 80
So if Shen drives fewer than 80 miles, the first plan will be cheaper. If he drives more than 80 miles, the second plan will be cheaper.
It's worth noting that this assumes that Shen is only considering the cost of the rental when making his decision. If there are other factors he is considering, such as convenience or availability, he may choose a different plan even if it ends up being slightly more expensive.
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In a baseball game, a pop fly is hit, and its height in meters relative to time in seconds is modeled by the function h(t) = -4. 9t^2 + 8t + 1
The maximum height reached by the pop fly is approximately 3.27 meters.
How to find the maximum height reached by the pop fly?
The equation h(t) = -4.9t^2 + 8t + 1 models the height in meters of a pop fly hit in a baseball game as a function of time in seconds.
The coefficient of t^2 is negative (-4.9), which means that the graph of this function is a downward-facing parabola. This makes sense, as the ball will start at a certain height and then be pulled down by gravity as it moves through the air.
The coefficient of t is positive (8), which means that the height of the ball is increasing at first. This makes sense, as the ball is gaining altitude after being hit.
The constant term (1) represents the initial height of the ball when it was hit.
To find the maximum height reached by the pop fly, we can find the vertex of the parabola. The x-coordinate of the vertex is given by -b/2a, where a is the coefficient of t^2 and b is the coefficient of t. In this case, a = -4.9 and b = 8, so the x-coordinate of the vertex is:
x = -b/2a = -8/(2*(-4.9)) = 0.8163
To find the corresponding y-coordinate, we can plug this value of t into the equation:
h(0.8163) = -4.9(0.8163)^2 + 8(0.8163) + 1 = 3.27
Therefore, the maximum height reached by the pop fly is approximately 3.27 meters.
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Charity can make 36 cupcake in 45 minutes. If she continues at this rate, how many cupcakes can she make in 8 hours?
a. 280 cupcakes b. 384 cupcakes c. 360 cupcakes d. 300 cupcakes
The total number of cupcakes charity can make in 8 hours is 384
The total number of cupcakes she can make in 45 minutes is 36
Cupcakes she can make in 1 minute = 36/45
Cupcakes she can make in 1 minute = 0.8
Cupcakes she can make in 8 hours
We will convert hours into minutes
1 hour = 60 min
8 hour = 8 × 60 min
8 hour = 480 min
Cupcakes she can make in 8 hours that is 480 min = 480 × 0.8
Cupcakes she can make in 8 hours = 384
Total number of cupcakes she can make is 384
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Find the area of an equilateral triangle with apothem length .
if necessary, write your answer in simplified radical form.
The area of an equilateral triangle with apothem length 'a' is (1/4) * √3 * [tex]a^2[/tex].
How to find the area of an equilateral triangle?Let's label the equilateral triangle as ABC. An apothem is a line segment from the center of the triangle to the midpoint of one of its sides, forming a right angle with that side. Let the apothem of the triangle be 'a'.
The apothem divides the equilateral triangle into two congruent 30-60-90 triangles, where the apothem is the hypotenuse of one of the triangles. The length of the apothem 'a' is also the height of each 30-60-90 triangle.
In a 30-60-90 triangle, the hypotenuse is twice the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg. So, the length of the shorter leg is a/2, and the length of the longer leg (which is also the length of one side of the equilateral triangle) is √3 times the length of the shorter leg. Thus, the length of one side of the equilateral triangle is:
s = √3 * (a/2) = (√3 / 2) * a
The area of the equilateral triangle can be calculated using the formula:
A = (1/2) * base * height
where the base is one side of the equilateral triangle, and the height is the length of the apothem 'a'. Substituting the values we found, we get:
A = (1/2) * s * a = (1/2) * (√3 / 2) * a * a = (1/4) * √3 *[tex]a^2[/tex]
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Select the correct answer. team goal scored in first five minutes p 2.34% q 3.56% r 1.24% s 4.01% t 3.88% total 2.86% the probabilities of a particular soccer team scoring a goal within the first five minutes of the game are given in the table. what is the probability of a goal being scored in the first five minutes of the game, given that the team is team q? a. 1.24% b. 2.86% c. 3.56% d. insufficient data
The probability of a goal being scored in the first five minutes of the game, given that the team is team q is 1.24%. The correct option is a.
The probability of a goal being scored in the first five minutes of the game, given that the team is team q, is given by the conditional probability:
P(goal scored in first 5 min | team is q) = P(goal scored in first 5 min and team is q) / P(team is q)
From the table, we have:
P(goal scored in first 5 min and team is q) = 3.56%
P(team is q) = 3.56%
Therefore:
P(goal scored in first 5 min | team is q) = 3.56% / 3.56% = 1
This means that if we know the team is team q, the probability of a goal being scored in the first five minutes of the game is 100% (or certain). So the correct answer is (a) 1.24%.
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_______ assisted Anton Raphael Mengs with the iconography of his ceiling fresco, Parnasus, in the Villa Albani.
A) Johann Winckelmann
B) Cardinal Albani
C) Jacques Louis David
D) Joshua Reynolds
David wants to buy a new bicycle that cost $295 before a 40% discount. He finds the cost
after the discount, in dollars, by evaluating 295 - 295(0. 40). His brother Michael finds the
same cost by evaluating 295(1 - 0. 40). What property can be used to justify that these two
expressions represent the same cost after the discount?
The expressions represent the same cost after the discount of 40%.
How to show that the two expressions 295 - 295(0.40) and 295(1 - 0.40) represent the same cost after the discount?
To show that the two expressions 295 - 295(0.40) and 295(1 - 0.40) represent the same cost after the discount, we can use the distributive property of multiplication over addition or subtraction.
The distributive property states that for any real numbers a, b, and c:
a(b + c) = ab + ac
a(b - c) = ab - ac
So, we can apply the distributive property as follows:
295 - 295(0.40)
= 295(1) - 295(0.40) [Multiplying 295 by 1]
= 295(1 - 0.40) [Using the distributive property]
Therefore, both expressions represent the same cost after the discount of 40%.
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An aquarium is 25 inches long, 12 1 half inches wide, and 12 3 over 4 inches tall. what is the volume of the aquarium?
hint: v= lwh
volume = length x width x height
Answer is: 3,984.375 cubic inches
To help you calculate the volume of the aquarium. Using the formula
V = L x W x H, where V is volume, L is length, W is width, and H is height:
Length (L) = 25 inches
Width (W) = 12.5 inches (12 + 0.5)
Height (H) = 12.75 inches (12 + 3/4)
Now, plug these values into the formula:
Volume (V) = 25 x 12.5 x 12.75
V = 3,984.375 cubic inches
The volume of the aquarium is 3,984.375 cubic inches.
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The length of a rectangle is 6 ft longer than its width. if the perimeter of the rectangle is 64 ft, find its length and width
The length of the rectangle is 19 feet and its width is 13 feet.
Let's denote the width of the rectangle by w. Then, according to the problem statement, the length of the rectangle is 6 feet longer, which means it is equal to w + 6.
The perimeter of a rectangle is given by the formula:
perimeter = 2 × length + 2 × width
Substituting the expressions for length and width that we have just found, we get:
64 = 2 × (w + 6) + 2w
Simplifying the right-hand side:
64 = 2w + 12 + 2w
64 = 4w + 12
52 = 4w
w = 13
So the width of the rectangle is 13 feet. Using the expression for the length we found earlier, the length is:
length = w + 6 = 13 + 6 = 19
Therefore, the length of the rectangle is 19 feet and its width is 13 feet.
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what is x^2-3x=70 in standard form?
Answer: x^2 + 3x - 70 = 0
Step-by-step explanation:
The probability that Mr Smith will have coffee with his breakfast is 0. 35. Find the probability that in the next 25 mornings, Mr Smith will have coffee on exactly 8 mornings
The probability that Mr Smith will have coffee on exactly 8 mornings out of the next 25 is 0.142, or 14.2%.
This scenario can be modeled by a binomial distribution, where:
The probability of success (having coffee) on any given morning is p = 0.35
The number of trials (mornings) is n = 25
The number of successes (mornings with coffee) we want to find the probability for is k = 8.
The probability mass function for a binomial distribution is given by:
[tex]P(X = k) = (n \: choose \: k) \times p^k \times (1-p)^{(n-k)},[/tex]
where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items out of n. It can be calculated as:
(n choose k) = n! / (k! × (n-k)!)
Using this formula and putting in the values we have,
[tex]P(X = 8) = (25 \: choose \: 8) \times 0.35^8 \times (1-0.35)^{(25-8)} [/tex]
[tex]P(X = 8) ≈ 0.142[/tex]
Therefore, the probability that Mr Smith will have coffee on exactly 8 mornings out of the next 25 is approximately 0.142, or 14.2%.
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A researcher collected the number of letters in each of 200 first names. The data are found to be normally distributed with a mean of 5. 82 and a standard deviation of 1. 43.
What percentage of first names have seven letters or less?
79. 4%
82. 5%
84. 1%
99. 8%
If a researcher collected the number of letters in each of 200 first names, approximately 79.4% of first names have seven letters or less. Therefore, the correct answer is 79.4%.
To find the percentage of first names with seven letters or less, we will use the mean (5.82) and standard deviation (1.43) of the normally distributed data. We will calculate the z-score for a name with seven letters:
z = (7 - 5.82) / 1.43
z ≈ 0.83
Now, using a z-table or a calculator that can compute the cumulative distribution function (CDF) of a standard normal distribution, we find the probability associated with the z-score:
P(z ≤ 0.83) ≈ 79.4%
So, approximately 79.4% of first names have seven letters or less. The correct answer is 79.4%.
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Are △abc and △def similar triangles? choose all that apply.
no, the corresponding sides are not proportional.
yes, the corresponding sides are proportional.
yes, the corresponding angles are all congruent.
no, the corresponding angles are not congruent.
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△ABC and △DEF are similar triangles if they have corresponding sides that are proportional and the corresponding angles are all congruent. Thus, the options that are applied are B and C.
Similar shapes are enlargements or shortening of other shapes using a scale factor.
Two triangles are said to be similar if the corresponding sides are proportional and the corresponding angles are the same. There are the following similarity criteria:
1. AA or AAA where all the angles are equal
2. SSS where all the sides are proportional to the corresponding sides
3. SAS where the corresponding sides and the angle between are proportional and congruent.
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Write the number in standard form. (8 × 10) + (1 × 1/10 ) + (6 × 1/1000 )
We can simplify the given expression first and then write it in standard form.
8 × 10 = 80
1 × 1/10 = 1/10
6 × 1/1000 = 6/1000 = 3/500
Adding these three values, we get:
80 + 1/10 + 3/500
To write this in standard form, we need to express it as a single number multiplied by a power of 10. We can do this by finding a common denominator for the fractions and adding them:
80 + 50/500 + 3/500 = 80 + 53/500
Now, we can write this as:
80.106
To express this in standard form, we move the decimal point to the left until there is only one non-zero digit to the left of the decimal point. We moved the decimal point 3 places to the left to get:
8.0106 × 10^1
Therefore, the number in standard form is 8.0106 × 10^1.