The maximal area of a right triangle with hypotenuse of length 3 is 0.
How to calculate the maximal area?Let's call the legs of the right triangle a and b.
Since the hypotenuse has length 3, we know that a² + b² = 3² = 9
The area of the triangle is given by A = 1/2 ab.
We want to maximize A subject to the constraint a² + b²= 9.
One way to do this is to use the method of Lagrange multipliers.
We want to maximize the function f(a, b) = 1/2 ab subject to the constraint g(a, b) = a²+ b² - 9= 0. The Lagrangian is then:
L(a, b, λ) = f(a, b) - λg(a, b) = 1/2 ab - λ(a² + b² - 9)
To find the maximum, we need to solve the system of equations:
∂L/∂a = 0∂L/∂b = 0∂L/∂λ = 0Taking the partial derivatives and setting them equal to zero, we get:
b/2 - 2λa = 0a/2 - 2λb = 0a² + b²- 9 = 0Solving the first two equations for a and b in terms of λ and plugging them into the third equation,
we get:
4λ² + 1 = 0
Then from the constraint a² + b² = 9, we have a = ±3.
The area of the triangle in this case is A = 1/2 ab = 0.
Therefore, the maximal area of a right triangle with hypotenuse of length 3 is 0, and it is achieved when one of the legs has length 0.
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Devon's tennis coach says that 72% of Devon's serves are good serves. Devon thinks he has a higher proportion of good serves. To test this, 50 of his serves are randomly selected and 42 of them are good. To determine if these data provide convincing evidence that the proportion of Devon's serves that are good is greater than 72%, 100 trials of a simulation are conducted. Devon's hypotheses are H o p = 72% and H a p > 72%, where p = the true proportion of Devon's serves that are good. Based on the results of the simulation, the estimated P-value is 0. 6. Using a= 0. 05, what conclusion should Devon reach? Because the P-value of 0. 06 > a, Devon should reject Ha. There is convincing evidence that the proportion of serves that are good is more than 72%. Because the P-value of 0. 06 > a, Devon should reject H a. There is not convincing evidence that the proportion of serves that are good is more than 72% Because the P-value of 0. 06 > a Devon should fail to reject H o. There is convincing evidence that the proportion of serves that are good is more than 72%. Because the P-value of 0. 06 > a Devon should fail to reject H o. There is not convincing evidence that the proportion of serves that are good is more than 72%
There is no convincing evidence that the proportion of Devon's serves that are good is more than 72%. The data collected does not provide sufficient evidence to support Devon's claim that he has a higher proportion of good serves than what his coach stated.
Based on Devon's hypotheses, H₀ states that p = 72%, while Hₐ states that p > 72%, where p represents the true proportion of Devon's good serves. To test this, 50 of his serves are randomly selected, and 42 are good. A simulation is conducted with 100 trials, resulting in an estimated P-value of 0.06. The significance level (α) is set at 0.05.
In this case, the P-value (0.06) is greater than the significance level (0.05). According to the rules of hypothesis testing, we should fail to reject the null hypothesis (H₀) when the P-value is greater than the significance level. Therefore, Devon should fail to reject H₀.
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To the nearest hundredth, what is the value of x?
Use a trigonometric ratio to compute a distance
Therefore, to the nearest hundredth, the value of x is 42.31 units.
What is triangle?A triangle is a three-sided polygon, which is a closed shape made up of straight lines. It is one of the simplest geometric shapes and is used extensively in mathematics, science, and engineering. In a triangle, each side connects two vertices or corners, and each vertex is where two sides intersect. The three angles of a triangle always add up to 180 degrees, and the sum of the lengths of any two sides is always greater than the length of the third side. Triangles can be classified by the lengths of their sides and the sizes of their angles, which gives rise to different types such as equilateral, isosceles, scalene, acute, right, and obtuse triangles. Triangles have many applications, such as in geometry, trigonometry, physics, and engineering, and they are fundamental to understanding the properties of other shapes and mathematical concepts.
Here,
In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, for this triangle, we have:
sin(53°) = opposite / hypotenuse
sin(53°) = x / 53
To solve for x, we can rearrange the equation as follows:
x = 53 * sin(53°)
Using a calculator to evaluate sin(53°), we get:
sin(53°) = 0.7986 (rounded to four decimal places)
Substituting this value into the equation, we get:
x = 53 * 0.7986
x = 42.308 (rounded to two decimal places)
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A pair of dice is tossed. Find the probability that the sum on the 2 dice is 4, given that doubles are rolled. (Enter your probability as a fraction.)
Answer:
1/6
Step-by-step explanation:
GEOMETRY HELP COSINE, SINE TANGENT
please help y’all i have no idea what i am doing
I need help also please explain as you go a long.
Given the expression: 5x10 − 80x2
Part A: Rewrite the expression by factoring out the greatest common factor. (4 points)
Part B:Factor the entire expression completely. Show the steps of your work. (6 points)
The entire expression is factored completely as: 5x2(x4 + 4)(x2 + 2)(x2 - 2)
Part A:
To factor out the greatest common factor, we need to find the largest number that divides evenly into both terms. In this case, the greatest common factor is 5x2.
5x10 − 80x2
= 5x2 (x8 - 16)
Therefore, we can rewrite the expression as 5x2(x8 - 16).
Part B:
To factor the entire expression completely, we need to use the difference of squares formula, which states that:
a2 - b2 = (a + b)(a - b)
In this case, we can rewrite the expression as:
5x2(x8 - 16) = 5x2[(x4)2 - (4)2]
Notice that x8 can be rewritten as (x4)2, and 80 can be factored into 4 x 20, which gives us 16 when squared.
Using the difference of squares formula, we can factor the expression further:
5x2[(x4 + 4)(x4 - 4)]
The expression (x4 + 4) cannot be factored further, but (x4 - 4) can be factored using the difference of squares formula again:
5x2[(x4 + 4)(x2 + 2)(x2 - 2)]
Therefore, the entire expression is factored completely as: 5x2(x4 + 4)(x2 + 2)(x2 - 2)
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A gardener has a rectangular vegetable garden that is 2 feet longer than it is wide. The area of the garden is at
least 120 square feet.
Enter an inequality that represents all possible widths, w, in feet of the garden
This is the inequality that represents all possible widths, w, in feet of the garden is W^2 + 2W - 120 ≥ 0
The area of a rectangle is given by the formula A = L x W, where A is the area, L is the length, and W is the width. In this problem, we are given that the garden is rectangular and that the length is 2 feet longer than the width, so we can write L = W + 2.
We are also told that the area of the garden is at least 120 square feet, so we can write:
A = L x W ≥ 120
Substituting L = W + 2, we get:
(W + 2) x W ≥ 120
expanding the left side, we get:
W^2 + 2W ≥ 120
Rearranging, we get:
W^2 + 2W - 120 ≥ 0
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There's a roughly linear relationship between the length of someone's femur (the long leg-bone in your thigh) and their expected height. Within a certain population, this relationship can be expressed using the formula h=62. 6+2. 35fh=62. 6+2. 35f, where hh represents the expected height in centimeters and ff represents the length of the femur in centimeters. What is the meaning of the hh-value when f=49f=49?
For an individual with a femur length of 49 centimeters, we can expect their height to be approximately 177.15 centimeters.
When f=49, plugging it into the formula h=62.6+2.35f, we get h=62.6+2.35(49)=177.15.
This means that for an individual with a femur length of 49 centimeters, we would expect their height to be approximately 177.15 centimeters.
This provides an estimate of the individual's height based on the relationship between femur length and height indicated by the formula. It's important to note that this is an estimate and individual variation may exist.
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If a 1. 5-volt cell is to be completely recharged, each electrion must be supplied with a minimum energy of
To completely recharge a 1.5-volt cell, each electron in the cell must be
supplied with a minimum energy of 2.403 × 10^-19 joules.
To completely recharge a 1.5-volt cell, each electron in the cell must be
supplied with an energy greater than or equal to the potential difference of
the cell, which is 1.5 volts.
The minimum energy required to supply to each electron can be
calculated
using the formula:
E = qV
where E is the energy in joules (J), q is the charge of an electron (1.602 ×
10^-19 coulombs), and V is the potential difference in volts.
Substituting the given values, we get:
[tex]E = (1.602 × 10^-19 C) × (1.5 V)E = 2.403 × 10^-19 J[/tex]
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The table below shows the number of gold, silver and bronze medals won by some
countries in the 1988 Winter Olympic Games.
Work out the ratio of gold to silver to bronze medals won by Sweden.
Give your answer in its simplest form.
Country
Canada
Finland
Soviet Union
Sweden
Gold
0
4
11
4
Silver
2
1
9
0
Bronze
3
2
9
2
Step-by-step explanation:
It looks as though ( from your post) Sweden won 4 golds and 0 silver and 2 bronze medals
4:0:2 simplifies to 2 :0 : 1
One of the teachers at a school is chosen at random. The probability that this teacher is female is 3/5 There are 36 male teachers at the school
If the probability that this teacher is female is 3/5 , there are a total of 90 teachers at the school.
Let's denote the total number of teachers at the school as T. We know that the probability of choosing a female teacher is 3/5. Therefore, the probability of choosing a male teacher is 1 - 3/5 = 2/5.
We are also given that there are 36 male teachers at the school. We can use this information to set up an equation:
36/T = 2/5
To solve for T, we can cross-multiply:
36 x 5 = 2 x T
180 = 2T
T = 90
Therefore, there are a total of 90 teachers at the school.
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Complete question is:
One of the teachers at a school is chosen at random. The probability that this teacher is female is 3/5 There are 36 male teachers at the school. Work out the total number of teachers at the school.
A function is a rule that assingns each value of independent variable to exactly value of the dependent variable
A function is a rule that assingns each value of independent variable to exactly one value of the dependent variable.
A function is a mathematical concept that relates two sets of values, known as the domain and the range. The domain is the set of independent variables, while the range is the set of dependent variables. A function is a rule that assigns to each value in the domain exactly one value in the range.
For example, if we have a function f(x) = 2x + 3, the domain would be any possible value of x, and the range would be any possible value of 2x + 3. So if we put x = 2, then f(x) = 2(2) + 3 = 7. Therefore, the function assigns the value of 7 to the value of 2 in the domain.
Functions are used in various branches of mathematics, science, and engineering to model and analyze relationships between two or more variables. They are an important concept in calculus, where they are used to study rates of change and optimization problems.
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Question: Nora needs to cut some equal pieces of yarn for her Science project. The piece of yarn she has is 67. 6 inches long. Each piece of yarn must be 1. 3 inches in lenght. How many pieces of yarn will Nora have.
Nora will be able to cut 52 equal pieces of yarn for her Science project.
To find out how many equal pieces of yarn Nora can cut for her Science project, we need to divide the total length of the yarn by the length of each piece.
Total length of yarn: 67.6 inches
Length of each piece: 1.3 inches
Step 1: Divide the total length by the length of each piece.
67.6 inches ÷ 1.3 inches = 52
Nora will have 52 equal pieces of yarn, each 1.3 inches long, for her Science project.
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Problem
Yoshi is a basketball player who likes to practice by attempting the same three-point shot until he makes the shot. His past performance indicates that he has a
30
%
30%30, percent chance of making one of these shots. Let
X
XX represent the number of attempts it takes Yoshi to make the shot, and assume the results of each attempt are independent.
Is
X
XX a binomial variable? Why or why not?
Choose 1 answer:
Choose 1 answer:
(Choice A)
A
Each trial isn't being classified as a success or failure, so
X
XX is not a binomial variable.
(Choice B)
B
There is no fixed number of trials, so
X
XX is not a binomial variable.
(Choice C)
C
The trials are not independent, so
X
XX is not a binomial variable.
(Choice D)
D
This situation satisfies each of the conditions for a binomial variable, so
X
XX has a binomial distribution
Choice D is correct: This situation satisfies each of the conditions for a binomial variable, so X has a binomial distribution.
A random variable X is said to have a binomial distribution if it satisfies the following conditions:
The variable X represents the number of successes in a fixed number of independent trials.
Each trial has only two possible outcomes: success or failure.
The probability of success is constant for each trial.
The trials are independent.
In this case, Yoshi attempts the same three-point shot until he makes the shot, so the number of attempts is not fixed. However, each attempt can be classified as a success (if he makes the shot) or a failure (if he misses the shot), so the variable X represents the number of successes in a sequence of independent trials with only two possible outcomes. Also, the probability of success is constant for each attempt, and the attempts are independent, so all four conditions for a binomial distribution are satisfied. Therefore, X is a binomial variable.
Choice D is correct: This situation satisfies each of the conditions for a binomial variable, so X has a binomial distribution.
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The area of a rectangle park is 53 square mile. The length of the park is 87 mile. What is the width of the park? URGENTTT PLS ANSWER STEP BY STEP
A circle is circumscribed around a regular octagon with side lemgths of 10 feet. Another circle is inscribed inside the octagon. Find the area. Of the ring created by the two circles. Round the respective radii of the circles to two decimals before calculating the area
The area of the ring created by the two circles is approximately 1374.63 square feet.
Let's first find the radius of the circumscribed circle. We can draw a diagonal of the regular octagon, which will be twice the length of one of its sides, forming an isosceles triangle with two radii of the circle.
The angle at the center of the circle between two adjacent sides of the octagon will be 360 degrees divided by 8, or 45 degrees.
The angle at the top of the isosceles triangle will be half of that, or 22.5 degrees. Using trigonometry, we can find the radius of the circumscribed circle:
[tex]$\sin(22.5^\circ) = \frac{opposite}{hypotenuse}$$\sin(22.5^\circ) = \frac{10}{2r}$$r = \frac{10}{2\sin(22.5^\circ)} \approx 21.21$[/tex]
Next, we can find the radius of the inscribed circle. Drawing radii from the center of the octagon to the points where it touches the circle, we can form 8 congruent isosceles triangles, each with a base of length 10 and two equal legs.
The angle at the top of each triangle will be half of the central angle between two adjacent sides of the octagon, or 22.5 degrees. Using trigonometry again, we can find the length of each leg of the triangle:
[tex]$\tan(22.5^\circ) = \frac{opposite}{adjacent}$$\tan(22.5^\circ) = \frac{r'}{5}$$r' = 5\tan(22.5^\circ) \approx 2.93$[/tex]
Now we can calculate the area of the ring created by the two circles:
[tex]$A = \pi R^2 - \pi r'^2$$A = \pi (21.21)^2 - \pi (2.93)^2 \approx 1374.63$ square feetTherefore, the area of the ring created by the two circles is approximately 1374.63 square feet.\\\\\\\\[/tex]
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Think about your daily experience how is probability utilized in news papers, television, shows, and radio programs that interest you? What are your general impression of the ways in which probability is used in the print media and entertainment industry
Probability is frequently used in news reports to convey the possibility of an event occurring.
Generally, in my opinion, probability is used well in the media space.
How Probability is Utilized?Probability is frequently used in news reporting to demonstrate the likelihood of an event occurring. A news story, for example, might mention that there is a 50% chance of rain tomorrow. Similarly, sports writers may use probability to forecast the outcome of games and goals to be scored.
Overall, I feel probability is utilized fairly responsibly in the media and entertainment industries, with a focus on informing or entertaining audiences rather than misleading them. However, in some cases, such as political polling or advertising, the use of probability may be incorrect or exploited to influence audiences.
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"Apply any appropriate Testing Method to:
[infinity]X
n=1
(−1)narctan n
n^2"
To test the convergence of the given infinite series, we can use the Alternating Series Test. The series is in the form: Σ((-1)^n * (arctan(n)/n^2)), for n = 1 to infinity.
The Alternating Series Test requires two conditions to be met:
1. The absolute value of the terms in the series must be decreasing: |a_n+1| ≤ |a_n|.
2. The limit of the terms in the series as n approaches infinity must be zero: lim (n→∞) |a_n| = 0.
For the given series, let's check these conditions: 1.The absolute value of the terms: |arctan(n)/n^2|. Since arctan(n) increases with n and n^2 increases faster than arctan(n), the ratio (arctan(n)/n^2) decreases as n increases. Therefore, this condition is met.
2. Now, we need to check the limit: lim (n→∞) |arctan(n)/n^2|. As n approaches infinity, the arctan(n) approaches π/2, and n^2 approaches infinity.
Therefore, the limit is (π/2)/∞ = 0, so the second condition is also met. Since both conditions are met, the Alternating Series Test confirms that the given series converges.
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"Complete question"
Apply Any Appropriate Testing Method To: ∞X N=1 (−1)Narctan N N2
Apply any appropriate Testing Method to:
∞X
n=1
(−1)narctan n
n2
The slant height if the cone is 13 cm. What is the volume of a cone having a radius of 5 cm and a slant height of 13 cm.
Thus, the volume of cone for the given slant height and radius is found as: 314 cu. cm.
Explain about the slant height of cone:The distance from a cone's apex to its outer rim is referred to as the segment's slant height. It is corresponding to the hypotenuse's length of a right triangle that creates the cone.
Given data:
slant height of cone l = 13 cm
radius r = 5 cm
Let h be the height
So, using Pythagorean theorem, find height.
l² = h² + r²
h² = l² - r²
h²= 13² - 5²
h² = 169 - 25
h = 12 cm
volume of a cone = 1/3 *π*r²*h
volume of a cone = 1/3 *3.14*5²*12
volume of a cone = 3.14*25*4
volume of a cone = 314 cu. cm
Thus, the volume of cone for the given slant height and radius is found as: 314 cu. cm.
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Simplify rational ( trinomial only)
Simplify the following expression completely.
Answer: [tex]\frac{x-4}{x-7}[/tex]
Step-by-step explanation:
[tex]\frac{x^{2}-5x+4 }{x^{2} -8x+7}[/tex]
factor the top and bottom by finding numbers that multiply to the last term but add to middle
for top part:
-4 and -1 multiply to +4 and add to -5
for bottom part:
-7 and -1 multiply to to +7 and add to -8
Those are your factored numbers, put into factored form
[tex]\frac{(x-4)(x-1)}{(x-7)(x-1)}[/tex] now cross off same factors from top and bottom only
[tex]\frac{x-4}{x-7}[/tex]
Bus stops A, B, C, and D are on a straight road. The distance from A to D is exactly 1 km. The distance from B to C is 2 km. The distance from B to D is 3 km, the distance from A to B is 4 km, and the distance from C to D is 5 km. What is the distance between stops A and C? AC=_ km
The distance between bus stops A and C is exactly 1 km.
To find the distance between bus stops A and C, we can use the fact that the distance from A to D is 1 km and the distance from C to D is 5 km.
This means that the total distance from A to C, passing through D, is 6 km (1 km + 5 km).
However, we need to subtract the distance between B and D (3 km) and the distance between B and C (2 km) since we don't want to double count the stretch between B and D.
Therefore, the distance between A and C is 6 km - 3 km - 2 km = 1 km.
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Two similar cylinders have heights 6cm and 30cm. The volume of the smaller cylinder is 90cm3. What is the volume of the larger cylinder?
Answer:
Step-by-step explanation:
Since the two cylinders are similar, their corresponding dimensions (radius and height) are proportional. Let the radius of the smaller cylinder be r.
Then, we can write:
r / 6 = R / 30
where R is the radius of the larger cylinder.
Simplifying this equation, we get:
R = 5r
Now, we can use the formula for the volume of a cylinder to find the volume of the larger cylinder:
Volume of smaller cylinder = πr^2h = 90 cm^3
Volume of larger cylinder = πR^2H = π(5r)^2(30) = 750πr^2 cm^3
Substituting R = 5r, we get:
Volume of larger cylinder = 750πr^2 cm^3
Therefore, the volume of the larger cylinder is 750π times the volume of the smaller cylinder:
Volume of larger cylinder = 750π(90 cm^3) = 67,500π/ cm^3 (approx. 211,239.74 cm^3 rounded to five decimal places).
The sum of the measurement of angle p and angle s is 140°.
• the measurement in degrees of angle p is represented by the expression (5x + 30)°
• the measure of angle s is 80°
What is the value of x?
A)38
B)6
C)10
D)22
Answer:
x=6
Step-by-step explanation:
(5x+30)+80=140
5x+110=140
5x=30
x=6
answer: B
Find the zeros of each quadratic equation below by graphing.
Pls I need help
The zeros of the quadratic equation are as follows
1. y = -x²+ 6x - 5:
zeros: (1, 0) and (5, 0)
2. y = x² + 2x + 1:
zeros: (-1, 0).
3. y = -x²+ 8x - 17:
zeros: (0, 0) and (0, 0)
4. y = x² - 4:
zeros: (1, 0) and (5, 0)
What is zero of a quadratic equation?Zero in a quadratic equation are x values that make the equation equal to zero. In other words, they are the x-intercepts or roots of a quadratic function.
Using graphical method, a zero is the point of intersection of the curve with the x -axis and this is shown in the graph attached.
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Recent studies show that the number of three-legged frogs in a particular area is increasing due to exposure to chemical pollutants. The first set of data reported in 2000 estimates a population of 5000 three-legged frogs. Statistics show an annual increase of 15%. Let denote the number of three-legged frogs projected to inhabit this area in the year 2000N. How many three-legged frogs are projected to inhabit this area by 2009? Round to the nearest whole number
By 2009, it is projected that approximately 13,956 three-legged frogs will inhabit the area.
Recent studies have indicated a growing concern for the population of three-legged frogs in a specific area, as they have been exposed to chemical pollutants. In the year 2000, data estimated that there were about 5,000 three-legged frogs (N) in this area. With an annual increase of 15%, we can project the number of frogs in future years using the formula:
Future population = N * (1 + growth rate) ^ number of years
In this case, we want to determine the number of three-legged frogs in the area by 2009. To calculate this, we will use the given values:
Future population = 5,000 * (1 + 0.15) ^ (2009 - 2000)
Future population = 5,000 * (1.15)⁹
Future population ≈ 13,956
Therefore, by 2009, it is projected that approximately 13,956 three-legged frogs will inhabit the area, rounding to the nearest whole number. This increase in population highlights the potential ecological consequences of chemical pollutants on the environment and the need for further investigation and mitigation measures.
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If the cube is divided into two equal parts by a plane parallel to the face defined by vertices 2, 3, 6, and 7, what will be the area of the cross-section?
A.
48 sq cm
B.
256 sq cm
C.
16 sq cm
The area of the cross-section is 16 sq. cm. Thus, option C is the correct answer.
Vertices sides = 2, 3, 6, and 7
Divide part face = parallel to the face of vertices
It is given that a square face is present in the middle of the cube. The area of the cross-section of the cube results from the plane and cube intersection.
To find the distance between the square face of the cube and the length of the side:
distance = [tex]\sqrt{[(x^{2} - x1)^2 + (y^{2} - y1)^2 + (z^{2} - z1)^2]}[/tex]
we can use the coordinates of any two adjacent sides to find the distance.
distance = [tex]\sqrt{[(3-2)^2 + (3-2)^2 + (3-1)^2] }[/tex]
distance = [tex]\sqrt{11}[/tex]
To calculate the area of the face of the cube:
area = [tex]side^{2}[/tex]
area = [tex]\sqrt{(11)^2}[/tex]
area = 11
The area of the cross-section can be estimated as:
area = (1/2) x 11 x 4) + 5 vertices of plane
area = 16 sq. cm
Therefore we can infer that the area of the cross-section is 16 sq. cm
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A new car is purchased for 29,000 and over time it’s value depreciates by one half every 3. 5 years what is the value of the car 20 years after it was purchased to the nearest hundred dollars
The required answer is the nearest hundred dollars: $902.09 is approximately $900.
To find the value of the car 20 years after it was purchased, we can use the formula for exponential decay:
Value = Initial value * (1 - Depreciation rate) ^ (time elapsed / time for depreciation)
1. Determine the depreciation rate: The car's value depreciates by one half every 3.5 years, so the depreciation rate is 50% or 0.5.
Depreciation is a term that refers to two aspects of the same concept: first, the actual decrease of fair value of an asset, such as the decrease in value of factory equipment each year as it is used and wears, and second, the allocation in accounting statements of the original cost of the assets to periods in which the assets are used (depreciation with the matching principle).
Depreciation is thus the decrease in the value of assets and the method used to reallocate, or "write down" the cost of a tangible asset (such as equipment) over its useful life span
2. Calculate the number of depreciation periods: Since the car's value halves every 3.5 years, we need to find out how many 3.5-year periods are in 20 years. To do this, divide 20 by 3.5: 20 / 3.5 ≈ 5.71 periods.
3. Use the exponential decay formula:
Value = 29,000 * (1 - 0.5) ^ (5.71)
Value ≈ 29,000 * (0.5) ^ (5.71)
Value ≈ 29,000 * 0.0311
Value ≈ 902.09
4. Round the value to the nearest hundred dollars: $902.09 is approximately $900.
So, the value of the car 20 years after it was purchased is approximately $900.
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In this problem you will only find the models. But make sure that you show all work to support
your answers. Use function notation in your final answers.
Suppose two types of wire will be used to form the edges of a rectangle. The wire used
for the width costs $2.50 per foot and the wire used for the height costs $4.25 per foot. Express the total cost of building the rectangle out of wire as a function of the with if the
enclosed area must be 500 square feet.
To start, let's call the width of the rectangle "w" and the height "h". We know that the area must be 500 square feet, so we can write an equation:
w*h = 500
We can solve this equation for h:
h = 500/w
Now we can express the total cost of the wire in terms of w. The cost of the wire for the width is $2.50 per foot, so the cost for that side is:
2.5w
The cost of the wire for the height is $4.25 per foot, so the cost for that side is:
4.25h = 4.25(500/w) = 2125/w
So the total cost of the wire is:
C(w) = 2.5w + 2125/w
This is our final answer expressed in function notation.
Let's denote the width of the rectangle as w and the height as h. We are given that the area of the rectangle must be 500 square feet, so we have:
w * h = 500
Now, we need to find the cost function based on the width. The cost of the wire for the width is $2.50 per foot and for the height is $4.25 per foot. Therefore, the total cost (C) can be expressed as:
C(w) = 2.50 * w + 4.25 * h
We need to express the height (h) in terms of the width (w) using the area equation:
h = 500 / w
Now, we can substitute this expression for h in the cost function:
C(w) = 2.50 * w + 4.25 * (500 / w)
This is the cost function for building the rectangle out of wire as a function of its width, given that the enclosed area must be 500 square feet.
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A point is dilated by a scale factor of 1/3 centered about the origin resulting in the new coordinates (-6,3). what are the coordinates of the point prior to the dilation
The coordinates of the point prior to the dilation are (-2,-1) when the Scale factor is 1/3 and the new coordinates are (-6,3).
To find the coordinates of the point prior to the dilation, we need to use the formula for dilation:
(x’, y’) = (k x, ky)
where
(x’, y’) = the new coordinates
(x, y) = original coordinates
k = scale factor
Given data:
Scale factor = 1/3
New coordinates = (-6, 3)
By substuting the values in the equation we get:
(-6, 3) = (k x, ky)
Solving for x and y:
k x = -6
ky = 3
Dividing the ky equation by the k x equation we get:
y/x = 3/-6
y/x = -1/2
From the above equation, we can assume that x = 2 and y = -1.
Therefore, the coordinates of the point prior to the dilation are (-2,-1).
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1. A tree was cut down. What 3D shape does it closely resemble?
A. Prism
B. Pyramid
C. Cylinder
D. Cone
2. A ball with radius 5. 5 cm fits tightly inside a cube. Find the volume of the
unoccupied space inside the cube. Round to the nearest cm.
A. 531
B. 166
C. 697
D. 634
The volume of the unoccupied space inside the cube is the volume of the cube minus the volume of the ball, which is approximately 166.
1. D. Cone. When a tree is cut down, its trunk typically has a roughly cylindrical shape with a tapered end, which closely resembles a cone.
2. B. 166. The diameter of the ball is 11 cm, which is also the length of the diagonal of the cube. Let's call the side length of the cube "s". Then, we can use the Pythagorean theorem to find s:
s^2 + s^2 + s^2 = 11^2
3s^2 = 121
s^2 = 121/3
The volume of the cube is s^3, which is approximately 166. The volume of the ball is (4/3)πr^3, where r is the radius of the ball. Since the ball fits tightly inside the cube, its diameter is equal to the side length of the cube, which is s√3. Thus, r = (s√3)/2 - 5.5.
The volume of the unoccupied space inside the cube is the volume of the cube minus the volume of the ball, which is approximately 166.
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10. Determine the line of regression and use it to find the best predicted new mileage rating of a ( point) four-wheel-drive vehicle given that the old rating is 19 mi/gal old 6 27 1 33 28 24 8 22 20 29 21 New 15 24 15 29 25 22 6 20 826 19 Oy 0.808 0.863x; 18.1 mi/gal Oy = 0.863 + 0.808x; 16.2 mi/gal oy 0.863 + 0.808x; 22.4 mi/gal y-0.808+ 0.863x; 17.2 mi/gal
The line of regression equation for the mileage rating of a ( point) four wheel drive vehicle is [tex]\hat y = 16.111 + 0.365x,[/tex] and the best predicted new mileage rating of a ( point) four-wheel drive vehicle when x = 19 mi/gal, is equals to the 23.046 mi/gal. So, option(b) is right one.
A linear regression line has an equation of the form [tex]\hat y = a + bx,[/tex]
where x is the independent variable and y is the dependent variable. The slope of the line is b, and a is the estimated intercept (the value of y when x = 0). We have a table form data of old and new rating of four-wheel-drive vehicles. We have to determine the line of regression. Now, we have to calculate the value of 'a' and 'b'. Let the old and new mileage rating of four-wheel-drive vehicles be represented by vaiables 'x' and 'y'. Using the following formulas, [tex]b =\frac{ S_{xy}}{S_{xx}}[/tex] where, [tex]S_{xx} = \sum x² - \frac{ (\sum x)² }{n} [/tex]
[tex]S_{xy} = \sum xy - \frac{ (\sum y \sum x) }{n}[/tex][tex]a = \bar y - b \bar x,[/tex]where , [tex]\bar x = \frac{\sum x }{n}[/tex]
[tex]\bar y = \frac{\sum y }{n}[/tex]Here, n = 11, [tex]\sum x[/tex] = 235
[tex]\sum xy[/tex] = 263, [tex]\sum x²[/tex] = 5733, [tex]\sum xy[/tex] = 5879, so
[tex]S_{xx}[/tex] = 5733 - (235)²/11
= 5733 - 5020.454 = 712.546
[tex]S_{xy}[/tex] = 5879 - (235×263)/11
= 260.364
Now, b = 260.364/712.546 = 0.365
a = (263/11) - 0.365 ( 235/11)
= 23.909 - 7.798
= 16.111
So, regression line equation is
[tex]\hat y = 16.111 + 0.365x,[/tex]
The best predicted value of y, when x = 19 mi/gal, [tex]\hat y = 19× 0.365 + 16.111[/tex]
=23.046 mi/gal
Hence, the best predicted value is
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Complete question:
10. Determine the line of regression and use it to find the best predicted new mileage rating of a ( point) four-wheel-drive vehicle given that the old rating is 19 mi/gal
old 6 27 17 33 28 24 8 22 20 29 21
New 15 24 15 29 25 22 6 20 82 6 19
a) y cap = 0.863 + 0.808x; 16.2 mi/gal
b) y cap = 16.111 + 0.365x; 23.04 mi/gal
c) y cap =0.808+ 0.863x; 17.2 mi/gal
d) y cap = 0.808 0.863x; 18.1 mi/gal