Optimization Exercises

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Here we’ll practice solving optimization problems.

Find the dimensions of the rectangle of largest area having fixed perimeter $100$ .

$\answer {25}\times \answer {25}$

Find the dimensions of the rectangle of smallest perimeter with fixed area $100$ . $\answer {10}\times \answer {10}$

A box with square base and no top is to hold a volume $100$ . Find the dimensions of the box that requires the least material for the five sides.

$\text {width}=\answer {2\cdot 5^{2/3}},\qquad \text {length}=\answer {2\cdot 5^{2/3}},\qquad \text {height}=\answer {5^{2/3}}$

A box with square base is to hold a volume $200$ . The bottom and top are formed by folding in flaps from all four sides, so that the bottom and top consist of two layers of cardboard. Find the dimensions of the box that requires the least material.

$\text {width}=\answer {100^{1/3}},\qquad \text {length}=\answer {100^{1/3}},\qquad \text {height}=\answer {2\cdot 100^{1/3}}$

A box with square base and no top is to hold a volume $V$ . Find (in terms of $V$ ) the dimensions of the box that requires the least material for the five sides.

$\text {width}=\answer {2^{1/3}V^{1/3}},\qquad \text {length}=\answer {2^{1/3}V^{1/3}},\qquad \text {height}=\answer {V^{1/3}/2^{2/3}}$

You have $100$ feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area?

$\answer {1250} square feet$

Marketing tells you that if you set the price of an item at $10 then you will be unable to sell it, but that you can sell 500 items for each dollar below $10 that you set the price. Suppose your fixed costs total $3000, and your marginal cost is $2 per item. What is the most profit you can make?

Find the area of the largest rectangle that fits inside a semicircle of radius $10$ (one side of the rectangle is along the diameter of the semicircle).

$\answer {100}$

A rectangle is inscribed in the ellipse $\frac {x^2}{3}+\frac {y^2}{5}=1$ Find the dimensions of the rectangle with the largest area. $\begin{align*} \text{base} &= \answer{\sqrt{6}}\\ \text{height} &= \answer{\sqrt{10}} \end{align*}$

What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere?

$\answer {1/\sqrt {3}}$

A trough is constructed with the following dimensions:

Find $x$ that maximizes the volume.

$x = \answer {2\sqrt {2}}$

Find the area of the rectangle with the largest area that fits beneath the curve $y=\sqrt {x}$ on the interval $[0,4]$ .

$A = \answer {16/(3\cdot \sqrt {3})}$