You have 100 meters of fencing material to enclose a rectangular plot. Your goal

Question

You have 100 meters of fencing material to enclose a rectangular plot. Your goal is to detremine the dimensions of the plot such that you enclose the maximum are possible.

a) Express the area function
ANSWER: A(x)= x(50-x)
b) What do you think the dimensions of the plot should be to enclose the maximum area? Explain
ANSWER: The dimensions of the plot should be 25x25 to enclose the maximum area possible given 100 meters of fencing.
c)Domain of the function/The value of x that yields the maximum area using a graphing calculator.
ANSWER: The Domain of the function is all real numbers. The value of X that yields the maximum are is 25.
d)Tell me how A(x)= 625-(x-25)^2 allows you to find the dimensions that produce a maximum area?
ANSWER: ?????????????????????
e)Discuss the strengths and weaknesses of the three strategies used in parts (b),(c) and (d)
ANSWER:?????????????????????????

Explanation / Answer

Answers to A, C and D If there is 100 meters of fencing, then one width and one length must equal half of that, 50 meters. So if the length is x meters, then the width is 50 - x meters.

(a)express the area A(x) of the rectangular plot as a function of the length x of one side.(the length)

A(x) = x(50 - x)

(c)what s the domain of the function?

Since 50 - x and x must be positive, 0 < x < 50. This is the domain.


(d) solve the problem analytically by showing that the area function can be written as A(x) = 625 - (x - 25)^2.how does this form of the function allow you to find the dimensions that produce a maximum area?

A(x) = x(50 - x)
A(x) = 50x - x²

the axis of symmetry includes the point that gives the maximum area. It is located on the axis of symmetry, which is found by the formula x = -b/(2a). In this case x = -50/(2*-1) = 25. This shows the maximum area occurs when the length is 25 meters. Since the width is 50 - x, it is also 25 meters. This gives a maximum area of 625 sq meters.

This also means the vertex of the graph is at (25,625)

The equation of a parabola in vertex form is y = a(x - h)² + k where (h,k) is the vertex and h = 25 and k = 625.

y = 625 - (x - 25)² is the function in vertex form.

I hope that helps!! :-) If there is 100 meters of fencing, then one width and one length must equal half of that, 50 meters. So if the length is x meters, then the width is 50 - x meters.

(a)express the area A(x) of the rectangular plot as a function of the length x of one side.(the length)

A(x) = x(50 - x)

(c)what s the domain of the function?

Since 50 - x and x must be positive, 0 < x < 50. This is the domain.


(d) solve the problem analytically by showing that the area function can be written as A(x) = 625 - (x - 25)^2.how does this form of the function allow you to find the dimensions that produce a maximum area?

A(x) = x(50 - x)
A(x) = 50x - x²

the axis of symmetry includes the point that gives the maximum area. It is located on the axis of symmetry, which is found by the formula x = -b/(2a). In this case x = -50/(2*-1) = 25. This shows the maximum area occurs when the length is 25 meters. Since the width is 50 - x, it is also 25 meters. This gives a maximum area of 625 sq meters.

This also means the vertex of the graph is at (25,625)

The equation of a parabola in vertex form is y = a(x - h)² + k where (h,k) is the vertex and h = 25 and k = 625.

y = 625 - (x - 25)² is the function in vertex form.

I hope that helps!! :-)

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