Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 70 feet of fencing to

Question

2 years ago

5

go around it, what dimensions will give him the maximum area in the garden? Round the answers to the nearest tenth.

1 answer:

viktelen [127] · Answer 1 · 2023-11-06T10:00:17+03:00

The dimension that would give the maximum area is 20.8569

<h3>How to solve for the maximum area</h3>

Let the shorter side be = x

Perimeter of the semi-circle is πx

Twice the Length of the longer side

$[70-(\pi )x -x]$

Length = $[70-(1+\pi )x]/2$

Total area =

area of rectangle + area of the semi-circle.

Total area =

$x[[70-(1+\pi )x]/2] + [(\pi )(x/2)^2]/2$

When we square it we would have

$70x +[(\pi /4)-(1+\pi)]x^2$

This gives

$70x - [3.3562]x^2$

From here we divide by 2

$35x - 1.6781x^2$

The maximum side would be at

$x = 35/2*1.6781$

This gives us 20.8569

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