The equation of the line AB,with slope of -1/2 is y = (-1/2)x + 7/2. We then use the dot product to show that the direction vectors of the lines AB and AC are orthogonal, and the coordinates of point C is (6,7).
In an orthonormal frame (o; i; j), the points A and B are given as A(3;1) and B(1;2), and the line (AC) is given by the equation 2x-y-5=0.
To determine the equation of line (AB), we need to find the slope (or gradient) of the line passing through A and B. We can find this by using the formula
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of two points on the line. Substituting the values of A and B, we get
slope = (2 - 1) / (1 - 3) = -1/2
Now that we have the slope, we can use the point-slope form of a line to find the equation of line (AB). Let (x,y) be any point on the line, then we have
y - 1 = (-1/2)(x - 3)
Simplifying, we get
y = (-1/2)x + 7/2
So the equation of line (AB) is y = (-1/2)x + 7/2.
To show that the direction vectors AB and AC of the lines (AB) and (AC) are orthogonal, we need to find the direction vectors of these lines and show that their dot product is zero. The direction vector of a line is just the vector that "points" in the direction of the line. To find it, we can take any two points on the line and subtract their coordinates to get a vector.
For line (AC), we can take A and C as two points on the line. Since we don't know the coordinates of C yet, we can solve the equation of line (AC) for x to get
x = (y + 5) / 2
Now substituting x by the above expression in the coordinates of point A we get
C((y + 5)/2, y)
Thus, the direction vector of line (AC) is
AC = C - A = ((y + 5)/2 - 3, y - 1) = (y/2 - 7/2, y - 1)
Similarly, we can take A and B as two points on the line (AB) to get the direction vector of line (AB):
AB = B - A = (1-3, 2-1) = (-2, 1)
Now, the dot product of AB and AC is
AB · AC = (-2)(y/2 - 7/2) + (1)(y - 1) = -y + 7
We can see that the dot product is zero when y=7. Hence, direction vectors AB and AC are orthogonal at point C(3,7).
To determine the coordinates of C, we substitute y=7 in the equation of line (AC) we obtained earlier
2x - y - 5 = 0
2x - 7 - 5 = 0
2x = 12
x = 6
Therefore, C has coordinates (6,7).
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Draw a line representing the "rise" and a line representing the "run" of the line. State
the slope of the line in simplest form.
Click twice to plot each segment.
Click a segment to delete it.
-10 -9 -8 -9 -6 -5 -4 -3
Slope of the Line:
-2
y
9
GO
5
St
3
N
M
24
RZ
-3
In
VS
09
F
2 3
Submit Answer
4
5 6 7 8 9
10
X
attempt 1 out of 2
Answer:
Step-by-step explanation:
A study found that 9% of dog owners brush the dogs teeth of 578 tall governors about how many would be expected to brush their dogs teeth?
Answer:
$\boxed{52}$ dog owners would be expected to brush their dog's teeth.
Step-by-step explanation:
NO LINKS!! URGENT HELP PLEASE!!
This prism has a right angle for a base.
What is the volume of the prism?
Explain your thinking
Answer:
48 unit cube
Step-by-step explanation:
Solution Given:
For Triangle ABC
hypotenuse AB= 5
base AC=4
By using hypotenuse law
height BC=[tex]\sqrt{5^2-4^2} =3[/tex]
Now Area of the right-angled triangle is 1/2 * base*height
=1/2*AC*BC=1/2*4*3=6 units square
As we know that
The volume of triangular prism: Area of base * height
over here Area of the base= Area of the Triangle and the height is the length of the prism
=6 units square*CF
=6 units square*8
=48 unit cube
Answer:
The volume of the given prism is 48 cubic units.
Step-by-step explanation:
The volume of a prism can be calculated by multiplying the area of its base by its height.
The bases of the given prism are the congruent right triangles ABC and DEF.
We have been given the measures of sides AB and AC of triangle ABC.
To determine the area of ΔABC, we first need to calculate the measure of side BC using Pythagoras Theorem:
[tex]\implies AC^2+BC^2=AB^2[/tex]
[tex]\implies 4^2+BC^2=5^2[/tex]
[tex]\implies 16+BC^2=25[/tex]
[tex]\implies BC^2=9[/tex]
[tex]\implies BC=3[/tex]
The area of a triangle is half the product of its base and height.
The base and height of a right triangle are its legs.
[tex]\begin{aligned}\textsf{Area of base}&=\textsf{Area of $\triangle ABC$}\\\\&=\dfrac{1}{2} \cdot BC \cdot AC\\\\&=\dfrac{1}{2} \cdot 3 \cdot 4\\\\&=6\; \sf square\;units\end{aligned}[/tex]
Therefore, the area of the base of the prism is 6 square units.
Finally, to find the volume of the prism, multiply the area of the base by the height of the prism:
[tex]\begin{aligned}\textsf{Volume of prism}&=\textsf{Area of base} \cdot \sf height\\\\&=\textsf{Area of $\triangle ABC$} \cdot \overline{CF}\\\\&=6 \cdot 8\\\\&=48\; \sf cubic\;units\end{aligned}[/tex]
Therefore, the volume of the given prism is 48 cubic units.
Find the square root of 1681
Answer: The square root of 1681 is 41.
Answer:
41
Step-by-step explanation:
It is 41
41 * 41 = 1681
41 ^ 2 = 1681
A bag contains pennies, nickels, dimes, and quarters. There are 50 coins in all. Of the coins, 12% are pennies and 34% are dimes. There are 3 more nickels than pennies. How much money does the bag contain?
Answer:
i think the bag contains $3.56
Step-by-step explanation:
If you want get 60000 in 8 years how much do you need to deposit in the bank today if the account pays an interest rate of 9%
Answer:
FV = PV x (1 + r)^n
where:
FV = future value
PV = present value (the amount to be deposited today)
r = interest rate per period (in this case, per year)
n = number of periods (in this case, 8 years)
Plugging in the given values, we get:
$60,000 = PV x (1 + 0.09)^8
Simplifying this equation:
PV = $60,000 / (1 + 0.09)^8
PV = $60,000 / 1.999^8
PV = $60,000 / 3.058
PV = $19,602.68
Therefore, the amount that needs to be deposited in the bank today to accumulate $60,000 in 8 years with an interest rate of 9% is $19,602.68.
I hope this helps.
the coordinate (-1,0) (1,0) and (0,1) form the vertices of a square true or false
Since the line splits the square into two regions with equal area, it goes through center of square.
Equation of line through point (12,12)
with slope 5
y−12=5(x−12)
y=5x−2
y-intercept=(0,−2)
Since line passes through points (0.4,0)
and (0.6,1)
it divides both the bottom and top of squares in the same ratio, to that the line divides the square into 2
trapezoids with height =1
and bases =0.4
and 0.6
. Therefore, they have the same area.
PLEASE HURRY
Let
u = 4i - j v = 3i + j w = i + 3j
Find the specified scalar.
u(v + w)
The specified scalar is 12 as the result of u(v + w) is 12i + 12j.
The provided vectors are given to be u = 4i - j v = 3i + j w = i + 3j. First, we need to find the vector v + w,
v + w = (3i + j) + (i + 3j)
v + w = 4i + 4j
Now, we can find dot product, u(v + w),
u(v + w) = u(4i + 4j)
u(v + w) = 4ui + 4uj
Substituting the given values of u, we get,
u(v + w) = 4(4i - j)i + 4(4i - j)j
u(v + w) = 16i - 4j + 16j - 4i
Simplifying, we get,
u(v + w) = 12i + 12j
Therefore, the specified scalar is 12.
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PLS HURRY I AM GIVING BRAINLIEST!!!
the question is in the photo!!
The sum of both sides is 2x² + 4. The length of the third side is 5x³ - 4x² + 3x - 12
What is a triangle?A triangle is a polygon that has three sides and three angles. Types of triangles are isosceles, right angled, scalene, equilateral.
Given that the perimeter of the triangle is 5x³ - 2x² + 3x - 8. The length of side one is 3x² - 4x - 1 and that of side two is -x² + 4x + 5
The sum of side one and side two is:
Sum = side 1 + side 2 = (3x² - 4x - 1) + (-x² + 4x + 5) = 2x² + 4
The sum of both sides is 2x² + 4
The third side is:
Third side = Perimeter - Sum of two sides = (5x³ - 2x² + 3x - 8) - (2x² + 4) = 5x³ - 4x² + 3x - 12
The length of the third side is 5x³ - 4x² + 3x - 12
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pleaseeee help me!!
mathematics the pic below
Answer:
15, 18, 21, 24, 27, 30 ( +3)
7, 14, 28, 56, 112, 224, 448 ( x 2)
77, 70, 63, 56, 49, 42, 35 ( - 7)
20, 31, 42, 53, 64, 75, 86, 97 (+11)
22, 35, 48, 61, 74, 87 (+13)
The graph of an equation is sketched in the figure.
Describe five ordered-pair solutions of this equation by
using a table.
X
-6
-3
0
3
6
y
☐
I
8-
10-88427
6
40
10
The value of five ordered-pair solutions of this equation by using a table are,
x y
- 6 - 1
- 3 - 2
0 3
3 - 4
6 - 5
We have to given that;
The graph of an equation is sketched in the figure.
Hence, From graph we get;
The value of five ordered-pair solutions of this equation by using a table are,
x y
- 6 - 1
- 3 - 2
0 3
3 - 4
6 - 5
Thus, The value of five ordered-pair solutions of this equation by using a table are,
x y
- 6 - 1
- 3 - 2
0 3
3 - 4
6 - 5
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Please help me with this
Answer: 1st option
explanation: The ones that are crossed out include 1 whole number and 7 1/8 pieces. this is equal to 1 and 7/8. Then there is 3 whole 1 pieces so the answer is the 1st option
I need help on this question
Answer:
C
Step-by-step explanation:
0 is base like for example negatives are below and positives are above and zero is the thing right in between so sea level would be 0!
Correct answer is 0.
Step-by-step explanation:Sea level si always the parameter to use when indicating the elevation of any place on earth. If you happen to be located at the same altitude than the sea, then you are at level 0 altitude in any unit of measure.
what is 5w–4w–w+6w simplified
The value of the expression 5w–4w–w+6w is 6w
What is simplification of expression?When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do.
For example, 2( 3a+5) can be simplified by using 2 to multiply both 3a and 5. This will give 6a+10. This means that the simplification of 2(3a+5) = 6a+10.
Similar , 5w-4w-w+6w can be simplified into:
5w+6w-4w-w
= 11w-5w
= 6w.
Therefore,the value of the expression 5w–4w–w+6w is 6w
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Helppp
The half-life of Radium-226 is 1590 years. If a sample contains 500 mg, how many mg will remain after 1000 years?---------
Using the exponential decay equation, we can see that after 1000 years we will have 178.43 mg of Ra₂₂₆
How much will remain after 1000 years?The decay of a radioactive substance, such as Radium-226, can be modeled by the exponential decay equation:
N(t) = N₀ * (1/2)^(t / T)
where:
N(t) is the amount of the substance remaining after time t
N₀ is the initial amount of the substance
t is the time elapsed
T is the half-life of the substance
Given that the half-life of Radium-226 is 1590 years and the initial amount is 500 mg, we can plug in these values into the equation and solve for N(1000), which represents the amount remaining after 1000 years.
N(1000) = 500 * (1/2)^(1000 / 1590)
N(1000) ≈ 178.43 mg
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what does scaled version mean in this case its 3:5 scaled version but i just want to know what it means asap please help
Scaled version means adjusted value from the original size. For example, 3:5 could be a scaled version of 6:10.
Understanding Scale VersionScale Version of an object is a replica or model of the original object that has been proportionally reduced or enlarged in size. The scale of a model is the ratio of the size of the model to the size of the original object. For example, if a model of a building is one-tenth the size of the actual building, then the scale of the model is 1:10, which means that every measurement on the model is one-tenth of the corresponding measurement on the actual building.
Scale versions of objects are often used in engineering, architecture, and design to create smaller or larger versions of objects that can be studied, tested, or used for other purposes. For example, architects might create scale models of buildings to study their design and appearance, while engineers might create scale models of machines or structures to test their performance under different conditions.
Scale versions can also be used in maps, where the scale is used to indicate the relationship between the size of the map and the size of the area it represents. In this case, the scale might be represented as a ratio, such as 1:10,000, or as a statement of the size of a unit of measurement on the map, such as 1 inch equals 1 mile.
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37. When making biscuits, a baker
mixes flour and sugar in the ratio
4:1. If he uses 6kg of sugar when
making some biscuits, how much
flour should he use?
(a) 30kg
(b) 24kg
C 10kg
D 8kg
E 1.5kg
The baker should use 24kg amount of flour when making the biscuits. The correct answer is (b) 24kg.
To determine the amount of flour the baker should use when making biscuits, we can use the given ratio of flour to sugar, which is 4:1.
Since the ratio is 4:1, for every 4 parts of flour, there is 1 part of sugar. Therefore, the ratio of flour to sugar can be expressed as 4/1.
If the baker uses 6kg of sugar, we can set up a proportion to find the corresponding amount of flour:
4/1 = x/6
Cross-multiplying, we get:
4 * 6 = 1 * x
24 = x
Therefore, the baker should use 24kg of flour when making the biscuits.
In summary, the correct answer is (b) 24kg.
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Jackson's Ice Cream Shop must have a mean of 110 visitors per day in order to make a profit. The table below shows the number of visitors during 1 week.
Day of the week | No of Visitors
Sunday.................... 63
Monday.................. 77
Tuesday.................. 121
Wednesday............... 96
Thursday .......................137
Friday................................ 154
Based on the table, how many visitors will need to go to Jackson's Ice Cream shop on Saturday in order for the company to make a profit for the week?
A-108
B-110
C-122
D-222
Please add work/ explain!!
To find out how many visitors Jackson's Ice Cream Shop needs on Saturday to make a profit for the week, we need to first calculate the total number of visitors for the six days already given in the table. Then we can find out the number of visitors needed on Saturday to reach the required mean of 110 visitors per day for the week.
Total number of visitors for the six days = 63 + 77 + 121 + 96 + 137 + 154 = 648
To make a profit for the week, the total number of visitors for the entire week needs to be:
110 visitors/day × 7 days/week = 770 visitors/week
We already have 648 visitors for the six days, so the number of visitors needed on Saturday to make a profit can be found by subtracting the total number of visitors already counted from the total number of visitors needed for the week:
770 - 648 = 122
Therefore, the answer is (C) 122 visitors on Saturday are needed to make a profit for the week.
what is the suface area of 11cm and 14cm
Answer:
429 i think
Step-by-step explanation:
Eight birds live in each bird house in Mrs.Tillman's yard. Complete the table to show how the number of birds,b, depends on the houses,h.
The number of birds in Mrs. Tillman's yard depends on the number of birdhouses in a linear fashion, with each birdhouse containing 8 birds
How to solveTo find the number of birds (b) in a given number of bird houses (h), we can use the formula: b = 8h
Now, we can complete the table:
Number of bird houses (h) Number of birds (b)
1 8
2 16
3 24
4 32
5 40
So, the number of birds in Mrs. Tillman's yard depends on the number of birdhouses in a linear fashion, with each birdhouse containing 8 birds.
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Mrs. Tillman has bird houses in her yard, with eight birds living in each bird house. Complete the table to show how the number of birds (b) depends on the number of bird houses (h):
Number of bird houses (h) Number of birds (b)
1
2
3
4
5
Can you guys help me out?
Step-by-step explanation:
For f(3) 3 is >= 2 so you use the second portion of the function :
f(3) = 5x+2 = 5(3) + 2 = 17
scheduled payment of $1010 due five months ago and $1280 due today are to be repaid by a payment of $615 in four month and the balance in seven months. If money is worth 7.75% p.a , and the focal date is in seven months, what is the amount of the final payment?
The amount of the final payment is $1,004.99.
To solve the given problem, we will use the concept of present value. We need to find the present value of the two payments due in the past, and then find the future value of the remaining payments due in the future.
Let's first find the present value of the two past payments:
PV1 = $1010 / (1 + 0.0775/12)⁵ = $868.15
PV2 = $1280 / (1 + 0.0775/12)⁰ = $1280
Next, we can find the future value of the remaining payments due in the future
FV1 = $615 x (1 + 0.0775/12)⁴ = $664.69
FV2 = X
Now we can set up an equation to solve for X, the final payment:
PV1 + PV2 = FV1 + FV2 / (1 + 0.0775/12)⁷
Substituting the values we have calculated:
$868.15 + $1280 = $664.69 + X / (1 + 0.0775/12)⁷
Simplifying and solving for X
X = ($868.15 + $1280 - $664.69) x (1 + 0.0775/12)⁷ = $1,004.99
Therefore, the amount of the final payment is $1,004.99.
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50 Points! Multiple choice algebra question. Suppose you deposit $1000 in an account paying 4% annual interest, compounded continuously. Find the balance after 10 years. Photo attached. Thank you!
The balance after 10 years is equal to: A. $1491.82.
How to determine the balance after 10 years?In Mathematics and Financial accounting, continuous compounding interest can be determined or calculated by using this mathematical equation (formula):
[tex]A(t) = P_{0}e^{rt}[/tex]
Where:
A(t) represents the future value.P₀ represents the principal.r represents the interest rate.t represents the time measured in years.When time, t = 10 years, the future value can be calculated as follows:
[tex]A(t) = 1000e^{0.04 \times 10}\\\\A(t) = 1000e^{0.4}[/tex]
A(t) = 1000(1.49182469764)
A(t) = 1491.82469764 ≈ $1491.82.
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-2x-2y=14 slecet all the ordered pairs that are solutions
The ordered pairs of the solutions to the linear expression are В. (-7, 0) C. (0, -7) and F. (3, -4)
What are the ordered pairs of the linear expressionFrom the question, we have the following parameters that can be used in our computation:
The linear expression -2x - 2y =14
To determine the ordered pairs of the linear expression, we set x and y to the values in the list of options and check for the true equation
Using the above as a guide, we have the following:
А. (2, 3) В. (-7, 0) C. (0, -7)
-2(2) - 2(3) = 14
-10 = 14 --- false
-2(-7) - 2(0) = 14
14 = 14 --- true
-2(0) - 2(-7) = 14
14 = 14 --- true
D. (-7, 1) E. (-1, 7) F. (3, -4)
-2(-7) - 2(1) = 14
12 = 14 --- false
-2(-1) - 2(7) = 14
-12 = 14 --- false
2(3) - 2(-4) = 14
14 = 14 --- true
Hence, the ordered pairs of the linear expression are В. (-7, 0) C. (0, -7) and F. (3, -4)
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Complete question
Select all the ordered pairs that are solutions
-2x - 2y = 14
А. (2, 3) В. (-7, 0) C. (0, -7)
D. (-7, 1) E. (-1, 7) F. (3, -4)
Show 2 different ways to find the value of x. What do you think is the most efficient method? Explain why.
We can use trigonometric relations or Pythagorean's theorem, we will see that x = 8ft.
How to find the value of x?We can see that x is the hypotenuse of the triangle, and we know the length of one leg and the angle between them, then we can use the cosine trigonometric relation:
cos(60°) = 4ft/x
solving for x:
x = 4ft/cos(60°) = 8ft
Other way to find x is first find the other side and then use the pyhtagorean theorem, to get the other side y we need:
tan(60°) = y/4ft
y = 4ft*tan(60°)
Then using Pythagorean's theorem we get:
x = √( (4ft)² + (4ft*tan(60°))²)
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How do l do this please help me asap
Answer:
[tex] {z}^{2} - 10z + 9 = [/tex]
[tex](z - 1)(z - 9)[/tex]
Please help Thanks
there is a FRACTION symbol two variable symbols y and x
I need to write an equation
The required equation of line is y = 2x
What is equation of line?A straight line's equation is y=mx+c, where m denotes the gradient and c denotes the height at which the line crosses the y-intercept, also known as the y-axis.
According to question:Given points:
(1,2) and (2,4)
then
[tex]Slope of line = \frac{4-2}{2-1}= 2[/tex]
then
y-2 = 2(x-1)
y = 2x - 2 + 2
y = 2x
Thus, required equation of line is y = 2x
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Stephanie bought two rugs. One rug is 15 feet long, and the other rug is 142 feet long. Each rug has a width of 10 feet.
What is the total area in square feet of the two rugs?
A 175 ft²
B 155 ft²
C 147.5 ft²
D 302.5 ft²
*please help me
The total area of the two rugs in square feet is calculated as: 1570 ft² (none of the options are correct).
How to Find the Total Area of the Rug?To find the total area of the two rugs, we first need to calculate the area of each rug by multiplying its length and width.
Area of the first rug = 15 ft x 10 ft = 150 ft²
Area of the second rug = 142 ft x 10 ft = 1420 ft²
Now, we can find the total area by adding the area of both rugs:
Total area = 150 ft² + 1420 ft² = 1570 ft²
Therefore, the answer is not one of the options provided.
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Look at the box plot shown below. What is the minimum value of the data set?
A box plot labeled Volunteer Service Hours with the beginning of quartile 1 at 13, the beginning of quartile 2 at 16.5, the beginning of quartile 3 at 20, the beginning of quartile 4 at 23, and the end of quartile 4 at 27.
The minimum value of the data set is 10.5.
Box plots are an effective graphical tool used to display a summary of a dataset. They provide a visual representation of the minimum and maximum values, quartiles, and outliers of the data. They are commonly used in statistics and can provide valuable insights into a dataset.
Now, let's consider the box plot in question. The box plot is labeled "Volunteer Service Hours" and it has four quartiles, each marked by a horizontal line. The first quartile (Q1) is marked at 13, the second quartile (Q2) is marked at 16.5, the third quartile (Q3) is marked at 20, and the fourth quartile (Q4) is marked at 23, with the end of the box extending to 27.
The minimum value of the data set can be found at the lower whisker of the box plot.
The lower whisker represents the minimum value that is not an outlier.
In this box plot, we do not see any outliers, so the minimum value of the dataset is represented by the lower end of the whisker, which is located at 10.5.
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The questions I need answered are in the picture… help plsss
The graph of the linear equation y = (1/2)*x + 3 can be seen in the image at the end.
How to graph the linear equation?Here we want to graph the linear equation:
y = (1/2)*x + 3
To graph it we need to find a few points on the line, graph them on a coordinate axis and then draw a line that connects them.
The first point that we can use is the y-intercept, it is what we get when we evaluate in x = 0.
y = (1/2)*0 + 3
The point is (0, 3)
Then we can get other points by evaluating in different values of x, for example, if x = 2
y = (1/2)*2 + 3 = 1+ 3 =4
If x = 4
y = (1/2)*4 + 3 = 2 + 3 = 5
So we have the points (2, 4) and (4, 5)
Now just graph all these points on a cordinate axis and connect them with a line, you will get a graph like the one in the image below.
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