Using the PERIOD method, design an experiment that compares how four different tires perform. You decide to run an experiment that tests all four tires on dry, damp, and wet roads for decelerations
lerations of 1, 3, 5, 7, and 9 m/s². If each experiment is repeated two times, how many measurements are needed?
measurements are needed. (Type a whole number)
640 measurements are needed to compare the performance of four different tires on dry, damp, and wet roads for decelerations of 1, 3, 5, 7, and 9 m/s², with two measurements per experiment.
How did we get the value?The PERIOD method stands for:
P: Performance metric (Deceleration)
E: Environmental condition (Dry, Damp, Wet)
R: Replicates (Two measurements per experiment)
I: Inputs (Four tires)
O: Output (Measurement result)
D: Design (Full factorial)
The number of measurements can be calculated as:
Measurements = R * (number of inputs)^E * P
In this experiment, there are:
R = 2 replicates
E = 3 environmental conditions (Dry, Damp, Wet)
P = 5 different decelerations (1, 3, 5, 7, 9 m/s²)
Inputs = 4 tires
Therefore, the number of measurements can be calculated as:
Measurements = 2 * 4^3 * 5 = 2 * 64 * 5 = 640
So, 640 measurements are needed to compare the performance of four different tires on dry, damp, and wet roads for decelerations of 1, 3, 5, 7, and 9 m/s², with two measurements per experiment.
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What is the length of the third side, or c?
The length of the third side or c=30 feet.
What is length?Length is defined as the measurement of distance of an object from one end to the other.
The length of the sides of a triangle is given by ,
a=24 feet, b=18 feet. c=?.
By Pythagorean theorem,
“In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“.
[tex]c^{2}=a^{2} +b^{2}[/tex]
[tex]c^{2} = 24^{2} +18^2[/tex]
= 900
⇒c=30 feet.
Hence, the length of the third side or c=30 feet.
Question:
The given diagram forms the triangle with the length of the sides ,
a=24 feet, b=18 feet. Find the length of the side c.
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14 friends are saving money for a trip to the movies. 6 of the friends put $1.50 into the
pile every day, while others put $1.50 into the pile every even day, and the rest of the
friends put $1.50 into the pile every odd day. If $12 was put into the pile on February
3rd, how much money will be put into the pile on February 4th?
Answer:
UM I THINK ITS 1500
Step-by-step explanation:
, $18 will be put into the pile on February 4th.
What is expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation.
Given, 14 friends are saving money for a trip to the movies
On February 3rd, a total of $12 was put into the pile. Let's find out how many friends put money into the pile that day.
The 6 friends who put $1.50 into the pile every day contributed:
6 friends x $1.50/friend/day = $9/day
So, on February 3rd, they would have contributed $9.
Let's assume that there are x friends who put $1.50 into the pile every even day.
Since February 3rd was an odd day, none of these friends would have contributed on that day.
Similarly, let's assume that there are y friends who put $1.50 into the pile every odd day.
Since February 3rd was an odd day,
y friends would have contributed $1.50 each, for a total of $1.50y.
Therefore, we can set up an equation to solve for y:
$9 + $1.50y = $12
Simplifying this equation, we get:
$1.50y = $3
y = 2
So, there are 2 friends who put $1.50 into the pile every odd day.
Thus, friends who contribute on even day are = 14 -6 -2 = 6
So,
on 4th February total money collected = money added by the friends who adds money every day + money added by the friends who adds money on even days.
Thus,
on 4th February total money collected = 6*1.5 + 6*1.5 = 9 + 9 =18
Therefore, $18 will be put into the pile on February 4th.
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which of the following is always true for all probability density functions of continuous random variables? group of answer choices they are symmetrical they are bell-shaped the area under the curve is 1.0 they have the same height
True, for all probability density functions of continuous random variables that the area under the curve is 1.0, but they do not have to be symmetrical or bell-shaped, and they do not have the same height.
A continuous random variable is a random variable that can take any value within a certain range, such as time or distance. Probability density functions (PDFs) describe the probability distribution of continuous random variables, and they have some important properties that hold true for all of them.
The first property of a PDF is that the area under the curve is always equal to 1.0. This means that the total probability of all possible outcomes of the random variable is equal to 1.0.
The second property is that PDFs do not have to be symmetrical or bell-shaped. The shape of the PDF depends on the distribution of the data.
The third property is that PDFs do not have the same height. The height of the PDF depends on the distribution of the data and the range of values that the random variable can take. The height of the PDF represents the density of the probability distribution for a given value.
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Witch equation is equivalent to n+4=11
The Equation which is Equivalent to n+4=11 is (n + 4) x 2 = 11 x 2.
What is Equivalent Expression?Expressions that are equivalent do the same thing even when they have distinct appearances. When we enter the same value for the variable, two algebraic expressions that are equivalent have the same value.
Given:
n+4=11
Solving the Equation
n= 11-4
n= 7
1. (n + 4) x 2 = 11
2n + 8 = 11
2n = 3
n= 3/2
2. (n + 4) x 2 = 11 / 2
2n + 8 = 11/2
2n = -5/2
n = -5/4
3. (n+ 4) x2 = 11 x 4
2n+ 8 = 44
2n = 36
n= 18
4. (n + 4) x 2 = 11 x 2
2n+ 8 = 22
2n = 14
n= 7
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The Question attached here is missing the options which are as follow:
(n + 4) x 2 = 11
(n + 4) x 2 = 11 / 2
(n+ 4) x2 = 11 x 4
(n + 4) x 2 = 11 x 2
2π + 5π= 7π proposition or not?
The statement 2π + 5π = 7π is always a proposition.
What is a proposition?A proposition is a statement that is either right or wrong. Both results can not be possible for a statement. Only one result occurs at a time.
Clearly, the value of the addition 2π + 5π is 7π. So, the statement 2π + 5π = 7π is always true.
Therefore, the obtained answer is a proposition.
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Work out
a) √11 × √11
b) √5 × √5 x√√2 × √2
c) √6 × √6 × √6 x
Note: Please leave your answer in surd form when appropriate.
The result of the evaluation of each of the expression are as follows;
a). 11b). 10c). 6√6 x.What are the results of.the expressions evaluation?Recall that; √a × √a = a.
As evident in the task content; the results of the expressions are as follows;
a). √11 × √11 = 11.
b). √5 × √5 x √2 × √2 = 5 × 2 = 10.
c). √6 × √6 × √6 x = 6√6 x.
Ultimately, the results are as evaluated above.
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A flagpole casts a 17-foot shadow on the ground when the sun is at a 59° angle of
elevation. Which of the following expressions can be used to determine the height (h), in
feet, of the flagpole? (Assume the flagpole is perpendicular to the ground.)
A. cos59° =h/17
B. tan59° =h/17
C. sin59° =h/17
D. cot59° =h/17
The expression tan59° = h/17 can be used to determine the height (h), in feet, of the flagpole.
What is the right triangle?A right triangle is defined as a triangle in which one angle is a right angle or two sides are perpendicular.
The height of the flagpole is the opposite side of the right triangle formed by the flagpole, its shadow on the ground, and the line from the top of the flagpole to the end of the shadow.
The angle of elevation of the sun is opposite the height of the flagpole, and the length of the shadow is the adjacent side.
Therefore, the tan of the angle of elevation is equal to the height of the flagpole divided by the length of the shadow:
tan59° = h/17
Thus, the correct expression to determine the height of the flagpole is:
B. tan59° = h/17
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write the expression as a square of a monomial. 64x^6
The expression 64x^6 can be written as a square of a monomial by taking the square root of both sides of the expression:
√(64x^6) = √(64) * √(x^6) = 8 * x^3
So, the expression 64x^6 can be written as the square of the monomial 8x^3.
Answer: √(64x^6) = √(64) * √(x^6) = 8 * x^3 So, the expression 64x^6 can be written as the square of the monomial 8x^3. The exact thing that legoslucaswang had said but it's just a different way of saying it.
Determine whether ZY || VX If WY=5, YX 12.5, WV=15, and WZ=4. Justify your answer.
V
N
W
Y
X
4
→>>>
wv
wv equals x because the hypotenuse of rhe quadrilateral is congruent to the pathagoreum theorum
Answer: wv
Step-by-step explanation:
When comparing the machining techniques for stainless steel sheet material to those for aluminum alloy sheet, it is normally considered good practice to drill the stainless steel at a
answer choices
A-higher speed with less pressure applied to the drill.
B-lower speed with more pressure applied to the drill.
C-lower speed with less pressure applied to the drill.
When comparing the machining techniques for stainless steel sheet material to those for aluminum alloy sheets, it is normally considered good practice to drill the stainless steel at a lower speed with less pressure applied to the drill. Hence, option C is the required accurate answer.
Stainless steel is a much harder and tougher material compared to aluminum, which makes it more difficult to machine.
Drilling at a higher speed with more pressure applied to the drill can cause the drill to overheat and wear out faster, resulting in lower-quality holes and increased production time.
Whereas, drilling at a lower speed with less pressure applied to the drill helps to reduce the risk of overheating, allows the drill to maintain its sharpness, and helps to produce higher-quality holes.
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How much does the water in a 6-person hot tub weigh?
Answer: A six-person model typically weighs approximately 1,000 pounds empty or 6,000 pounds full.
can you plese do this
The vector is (-3, 4) and its x and y components are -3 and 4.
What are Vectors?A quantity or phenomena with independent qualities for both magnitude and direction is called a vector. The term can also refer to a quantity's mathematical or geometrical representation. Examples of vectors in nature are velocity, momentum, force, electromagnetic fields and weight.
Given:
Vector v have an initial point at (3, -5) and a terminal point at (0, -1).
It initial point of a vector is (a, b) and terminal point is (c, d), then the vector is
v= (c-a, d-b)
We have (a, b) = (3, -5) and (c, d) = (0, -1)
Then, vector v is defied as
v = (0 - 3, -1 - (-5))
v = (-3, -1+5)
v = (-3, 4)
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Let f(x) = 4x2 – 63. We want to estimate f(3.05) using linear approximations. That is, using an appropriate tangent line. First, we will build the tangent line at (x, y)=_____ Enter as an ordered pair (a,b). The slope of the tangent line comes from f'. For this problem, f'(x) = ______
And mtan = _____
The equation of the tangent line, in slope intercept form, is y = T(x) = ______
Now, f(3.05) = T(3.05) =______
Compare to actual value f(3.05) =_____
The tangent line at (x, y) = (3, 33) is y = 25x + 8.Enter as an ordered pair (a,b). The slope of the tangent line comes from f'. For this problem,f'(x) = 8x. And mtan = f'(3) = 24.The equation of the tangent line in slope-intercept form is y = T(x) = 24x - 69.
T(3.05) = 24(3.05) - 69 = -18.2.
f(3.05) = 4(3.05)^2 - 63 = -16.73.
We want to estimate the value of f(x) = 4x^2 - 63 at x = 3.05 using linear approximations. This means we need to find the equation of the tangent line to the graph of f at x = 3, which will give us a good approximation of f(3.05) near x = 3.
To find the tangent line, we first need to find the slope of the tangent line, which is given by the derivative of f at x = 3. We have f(x) = 4x^2 - 63, so f'(x) = 8x. Therefore, f'(3) = 24, which is the slope of the tangent line at x = 3.
Next, we need to find a point on the tangent line. We can use the point (x, y) = (3, 33), which is on the graph of f and also happens to be at x = 3. This means the tangent line at (3, 33) will be very close to the graph of f near x = 3.
Using the point-slope form of a line, we can find the equation of the tangent line at (3, 33):
y - 33 = 24(x - 3)
Simplifying this equation gives us the slope-intercept form of the tangent line:
y = 24x - 69
Now we can use this equation to estimate the value of f(3.05) by plugging in x = 3.05:
T(3.05) = 24(3.05) - 69 = -18.2
This means the tangent line predicts that f(3.05) is approximately -18.2.
To compare this to the actual value of f(3.05), we can plug it into the original function:
f(3.05) = 4(3.05)^2 - 63 = -16.73
So the actual value of f(3.05) is approximately -16.73, which is close to the estimate given by the tangent line.
In summary, using linear approximations, we found that the tangent line at (3, 33) has a slope of 24 and passes through the point (3, 33). The equation of the tangent line is y = 24x - 69. Using this equation, we estimated that f(3.05) is approximately -18.2, which is close to the actual value of -16.73.
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Which rational number is greatest in-10/11,-19/22,-23/33,-39/44
The Greatest Number is -10/11.
What is Ascending Order?Numbers can be arranged in ascending order, from least value to highest value. The arrangement is left to right.
Given:
Rational Number: 10/11,-19/22,-23/33,-39/44
The LCM of 11.22,33,44 is 132
-10/11×12/12=-120/132
-19/22×6/6=-114/132
-23/33×4/4=-92/132
-39/44×3/3=-117/132
In ascending order
-120/132,-117/132,-114/132,-92/132
= -10/11,-39/44,-19/22,-23/33
In descending order
-92/132,-114/132,-117/132,-120/132
= -23/33,-19/22,-39/44,-10/11
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Suppose that 20 randomly selected customers give the following satisfaction ratings (on a scale of 1 to 10) for a DVD recorder N 1 1 4 4 4 4 7 7 6 8 8 8 8 9 9 9 10 10 10 Find the first quartile, the median, and the third quartile for these data (Round your answers to 1 decimal place.), Q1 Median Q3
From the given information provided, the first quartile (Q₁) is 4, the median is 8, and the third quartile (Q₃) is 9.
To find the first quartile, median, and third quartile for the given data set, we first need to arrange the data in ascending order:
1 1 4 4 4 4 6 7 7 8 8 8 8 9 9 9 10 10 10
There are 20 data points in the set, so the median is the average of the 10th and 11th values:
Median = (8 + 8) / 2 = 8
To find the first quartile (Q₁), we need to find the median of the lower half of the data set. There are 10 data points in the lower half, so the median is the average of the 5th and 6th values:
Q₁ = (4 + 4) / 2 = 4
To find the third quartile (Q₃), we need to find the median of the upper half of the data set. There are 10 data points in the upper half, so the median is the average of the 15th and 16th values:
Q₃ = (9 + 9) / 2 = 9
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Decide how many solutions this equation has:
x2 + 3 = 0
Decide how many solutions this equation has:
x2 - 2x + 1 = 0
how many solutions this equation has:
x2 + 3 = 0
answer : no real solution
x = + i√3
x = - i√3
Decide how many solutions this equation has:
x2 - 2x + 1 = 0
answer : has 1 solution
x=1
chatgpt
What is the product of 2/5 and 4/5
Q9. A triangular prism has a volume of 480 cm³. The cross section of the prism is an isosceles triangle with base 6 cm. The depth of the prism is 20 cm. Find the length of the longest side of the triangular cross section. (Answer correct to 2 d.p.)
Answer:
Step 1: The volume of a prism is determined by the area of the cross section multiplied by the height of the prism.
Step 2: The area of an isosceles triangle can be calculated by A = (base x height)/2.
Step 3: Substituting values, we get A = (6 cm * h)/2.
Step 4: Since the volume of the triangular prism is 480 cm3, substituting in the formula V = A * height, we get 480 cm3 = (6 cm * h)/2 * 20 cm.
Step 5: Solving for h, we get h = 16 cm.
Step 6: Since it is an isosceles triangle, the longest side of the cross section is also 16 cm.
Answer: The length of the longest side of the triangular cross section is 16 cm.
if three letter of the alphabet are slected at random find the porbability of getting all vowels letters are relaced each time before selection
The probability of getting all vowels when three letters of the alphabet are selected at random with replacement is 1/125.
There are 5 vowels in the English alphabet (a, e, i, o, and u) and 26 letters in total. If we select three letters at random with replacement, we have 5 options for each selection.
The probability of getting a vowel on any one selection is 5/26. Since we are replacing the letter each time, the probability of getting three vowels in a row is:
(5/26) × (5/26) × (5/26) = 1/125
Therefore, the probability of getting all vowels when three letters of the alphabet are selected at random with replacement is 1/125.
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Referring to the figure, evaluate the expression shown
when a = 3, b = 7, c = 2
Answer:
5
Step-by-step explanation:
let fxn; n 0g be a dtmc on state space f1; 2; : : : ; ng. suppose it incurs a cost of c.i / dollars every time it visits state i. let g.i / be the total expected cost incurred by the dtmc until it visits state n starting from state i. derive the following equations: g.n / d0; g.i / dc.i / c pn jd1 pi;j g.j /; 1 j n 1
We can derive the equations using the principle of optimality for Markov decision processes. This principle states that an optimal policy for a process must satisfy the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
Let g(i) denote the total expected cost incurred by the DTMC until it visits state n starting from state i. We can express this as:
g(i) = c(i) + p(i, i+1)g(i+1) + p(i, i+2)g(i+2) + ... + p(i, n-1)g(n-1)
Here, p(i, j) denotes the transition probability from state i to state j. The first term c(i) represents the cost incurred by visiting state i for the first time, and the second term p(i, i+1)g(i+1) represents the expected cost of moving from state i to state i+1 and continuing optimally from state i+1. The same applies to the following terms, with the last term p(i, n-1)g(n-1) representing the expected cost of moving from state i to state n-1 and continuing optimally from state n-1 to state n.
To solve for g(n), we note that the expected cost to reach state n from state n is zero since we are already at the target state. Therefore, we have:
g(n) = 0
Now, we can use the principle of optimality to solve for the remaining g(i)'s. Specifically, we consider the expected cost of moving from state i to state i+1 and continuing optimally from state i+1. This can be expressed as:
c(i) + p(i, i+1)g(i+1)
The optimal decision is to minimize this cost, which can be achieved by choosing the state j that minimizes the cost of moving from state i to state j and continuing optimally from state j. Therefore, we have:
g(i) = c(i) + min_j{p(i, j)g(j)}, for 1 <= i < n
This equation states that the expected cost of visiting state i and continuing optimally to state n is the sum of the cost of visiting state i and the expected cost of moving from state i to the state j that minimizes the expected cost of moving from state i to state j and continuing optimally from state j.
Combining the two equations, we obtain the desired results:
g(n) = 0
g(i) = c(i) + min_j{p(i, j)g(j)}, for 1 <= i < n
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I need to complete the tables using the equation..
The values of y when the equation is y = 8x + 5 will be -3,5,13,21,29.
How to calculate the valuesAn equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.
Since y = 8x + 5
when x = -1, y will be:
= 8(-1) + 5
= -3
when x = 0, y will be:
= 8(0) + 5
= 5
when x = 1, y will be:
= 8(1) + 5
= 13
when x = 2, y will be:
= 8(2) + 5
= 21
when x = 3, y will be:
= 8(3) + 5
= 29
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5) h(x) = -2x² - 5
g(x) = 2x - 1
Find (h•g)(x)
The value of of function is -8x² + 8x - 7.
What is a function?
Function is a type of relation, or rule, that maps one input to specific single output.
In mathematics, a function is an expression, rule, or law that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) (the dependent variable). In mathematics and the physical sciences, functions are indispensable for formulating physical relationships.
Linear function is a function whose graph is a straight line
We are given that;
The function h(x) = -2x² - 5 and g(x) = 2x - 1
Now,
To find (h•g)(x), we need to first evaluate the composition of functions h(g(x)):
h(g(x)) = -2(2x - 1)² - 5 [substituting g(x) into h(x)]
= -2(4x² - 4x + 1) - 5
= -8x² + 8x - 7
Therefore, the function (h•g)(x) will be equal to -8x² + 8x - 7.
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Please help answer this math problem!
Answer:
357
Step-by-step explanation:
0.22x=49.95+0.08x
Answers basically x+1
The Garcia family drew a circle graph of their budget that contained the following: Taxes, 20% Rent, 32% Food, 20% Utilities, 5% Gas, 13% Miscellaneous, 12% What would you tell the family concerning the data?
If the Garcia family drew a circle graph of their budget I would tell the Garcia family that the circle graph shows the percentage breakdown of their budget.
What would you tell the family concerning the data?The largest portion of their budget is going towards rent, followed by taxes and food. Utilities, gas, and miscellaneous expenses make up smaller portions of their budget.
It may be helpful for the family to further analyze their spending in each category to see if any adjustments can be made to better align with their financial goals.
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One solution to a quadratic function, h, is given.
-4 + 7i
Which statement is true?
A.
Function h has no other solutions.
B.
The other solution to function h is -4 − 7i.
C.
The other solution to function h is4 − 7i.
D.
The other solution to function h is 4 + 7i.
The correct statement is the other solution to the quadratic function h is -4-7i.
What is a quadratic function?A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two.
Given that, one of the solution to a quadratic function, h, is given, -4 + 7i,
We know that,
The solution of a quadratic function, can be found by using the quadratic formula, which is given by,
x = -b±√b²-4ac / 2a
Here, ± is showing that there are two values of x one in positive and another in negative,
We have one solution = -4 + 7i
Therefore, another solution will be its opposite = -4 - 7i
Hence, the correct statement is the other solution to the quadratic function h is -4-7i.
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A function is defined by the equation f(x) = 2x - 5.
a. What is f(0)?
b. What is ƒ(1/2)?
c. What is f(100)?
d. What is x when f(x) = 9?
Answer:
a. f(0)= -5
b. f(1/2) = -4
c. f(100) = 195
d. x = 7
Step-by-step explanation:
Given that,
f(x) = 2x - 5 .........(1)
a)
Putting x = 0 in equation (1), we have
f(0) = 2(0) -5 = 0 - 5 = -5
b)
Putting x = 1/2 in equation (1), We have
f(1/2) = 2(1/2) -5 = 1 - 5 = -4
c)
Putting x = 100 in equation (1), We have
f(100) = 2(100) -5 = 200 - 5 = 195
d)
Putting f(x) = 9 in equation (1), We have
9 = 2x - 5
⇒ 2x = 9 + 5
⇒ 2x = 14
⇒ x = 14/2
⇒ x = 7
The tailgate of a truck is 2 feet above the ground. The incline of a ramp used for loading the truck is 11°, as shown below. 2' 11° Find, to the nearest tenth of a foot, the length of the ramp.
The required length of the ramp is approximately 10.4 feet.
What is trigonometry?Trigonometry is essentially the study of triangle calculations (hence the name trigonometry). It is a mathematical study of connections involving the lengths, heights, and angles of various triangles. The discipline was created in the third century BC as a result of the use of geometry in astronomical research.
According to question:Let's call the length of the ramp "x". We can then use the sine function to relate the angle and the height of the ramp:
sin(11°) = opposite/hypotenuse
sin(11°) = 2/x
We can then solve for x:
x = 2/sin(11°)
x ≈ 10.4 feet
Therefore, the length of the ramp is approximately 10.4 feet.
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A random rectangle is formed in the following way: The base, X, is chosen to be a uniform [0, 1] random variable and after having generated the base, the height is chosen to be uniform on [0, X]. Use the law of total expectation, Theorem A of Section 4.4.1, to find the expected circumference and area of the rectangle.
Using the law of total expectation, the expected circumference of the rectangle is 3/2, and the expected area of the rectangle is 1/6.
Let C and A denote the circumference and area of the rectangle, respectively. Then we have:
E[C] = E[C|X] * P(X) + E[C|X'] * P(X')
where X' is the complement of X, and P(X) and P(X') are the probabilities of X and X' respectively. Since X is a uniform [0, 1] random variable, we have P(X) = 1 and P(X') = 0.
Therefore, we can simplify the above expression to:
E[C] = E[C|X]
To find E[C|X], we can use the formula for the circumference of a rectangle:
C = 2 * (base + height)
Substituting in the values for the base and height of the rectangle, we get:
C = 2 * (X + U[0, X])
where U[0, X] denotes a uniform random variable on [0, X].
Then, we can find the expected value of C given X as follows:
E[C|X] = E[2*(X+U[0,X])|X]
= 2*(X+E[U[0,X]|X])
where we have used the linearity of expectation. Note that E[U[0,X]|X] is simply the expected value of a uniform random variable on [0, X], which is X/2.
Therefore, we have:
E[C|X] = 2*(X+X/2) = 3X
Substituting this back into the expression for E[C], we get:
E[C] = E[3X] = 3E[X]
Since X is a uniform [0, 1] random variable, we have E[X] = 1/2. Therefore, the expected circumference of the rectangle is:
E[C] = 3E[X] = 3/2
Similarly, we can use the Law of Total Expectation to find the expected area of the rectangle by conditioning on X:
E[A] = E[A|X] * P(X) + E[A|X'] * P(X')
where, again, we have P(X) = 1 and P(X') = 0. Therefore, we can simplify the expression to:
E[A] = E[A|X]
To find E[A|X], we can use the formula for the area of a rectangle:
A = base * height
Substituting in the values for the base and height of the rectangle, we get:
A = X * U[0, X]
where U[0, X] denotes a uniform random variable on [0, X].
Then, we can find the expected value of A given X as follows:
E[A|X] = E[XU[0,X]|X]
= XE[U[0,X]|X]
where we have again used the linearity of expectation. Note that E[U[0,X]|X] is X/2, as before.
Therefore, we have:
E[A|X] = X * (X/2) = X^2/2
Substituting this back into the expression for E[A], we get,
E[A] = E[X^2/2] = 1/2 * E[X^2]
To find E[X^2], we can use the formula for the variance of a uniform [0, 1] random variable:
Var(X) = E[X^2] - (E[X])^2 = 1/12
Solving for E[X^2], we get:
E[X^2] = Var(X) + (E[X])^2 = 1/12 + (1/2)^2 = 1/3
Substituting this back into the expression for E[A], we get:
E[A] = 1/2 * E[X^2] = 1/6
Therefore, the expected area of the rectangle is:
E[A] = 1/6
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