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Mekhanik [1.2K]
3 years ago
11

You have 42 ft. of fencing (1 ft. segments) to make a rectangular garden. How much should each side be to maximize your total ar

ea? What will the total area be for the garden? Find the length, width, and area of your garden that will maximize its total area.
Length:
Width:
Area:

Mathematics
1 answer:
Verizon [17]3 years ago
6 0

Answer:

Length=10.5\ ft

Width=10.5\ ft

Area=110.25\ ft^{2}

Step-by-step explanation:

Let

x----> the length of the rectangular garden

y---> the width of the rectangular garden

we know that

The perimeter of the rectangle is equal to

P=2(x+y)

we have

P=42\ ft

so

42=2(x+y)

simplify

21=(x+y)

y=21-x------> equation A

Remember that the area of rectangle is equal to

A=xy ----> equation B

substitute equation A in equation B

A=x(21-x)

A=21x-x^{2}----> this is a vertical parabola open downward

The vertex is a maximum

The y-coordinate of the vertex is the maximum area

The x-coordinate of the vertex is the length side of the rectangle that maximize the area

using a graphing tool

The vertex is the point (10.5,110.25)

see the attached figure

so

x=10.5\ ft

Find the value of y

y=21-10.5=10.5\ ft

The garden is a square

the area is equal to

A=(10.5)(10.5)=110.25\ ft^{2} ----> is equal to the y-coordinate of the vertex is correct

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Answer:

37.70% probability that the student will pass the test

Step-by-step explanation:

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Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

In which C_{n,x} is the number of different combinations of x objects from a set of n elements, given by the following formula.

C_{n,x} = \frac{n!}{x!(n-x)!}

And p is the probability of X happening.

10 true/false questions.

10 questions, so n = 10

True/false questions, 2 options, one of which is correct. So p = \frac{1}{2} = 0.5

If a student guesses on each question, what is the probability that the student will pass the test?

P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

P(X = 6) = C_{10,6}.(0.5)^{6}.(0.5)^{4} = 0.2051

P(X = 7) = C_{10,7}.(0.5)^{7}.(0.5)^{3} = 0.1172

P(X = 8) = C_{10,8}.(0.5)^{8}.(0.5)^{2} = 0.0439

P(X = 9) = C_{10,9}.(0.5)^{9}.(0.5)^{1} = 0.0098

P(X = 10) = C_{10,10}.(0.5)^{10}.(0.5)^{0} = 0.0010

P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) = 0.2051 + 0.1172 + 0.0439 + 0.0098 + 0.0010 = 0.3770

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