Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Measuring Change: Differentiation, Exercise 53: Exercise 5Q

A farmer wishes to enclose a rectangular field of area $100 {\mathrm{m}}^2$. The farmer must buy fencing material for three of the sides, but the fourth side (one of the longer sides) will be an existing fence. The shorter sides of the rectangular enclosure are to have length $x \mathrm{m}$.

Find the minimum length of fencing required, and the value of $x$ for which this minimum occurs.

A farmer wishes to enclose a rectangular field of area $100 {\mathrm{m}}^2$. The farmer must buy fencing material for three of the sides, but the fourth side (one of the longer sides) will be an existing fence. The shorter sides of the rectangular enclosure are to have length $x \mathrm{m}$.

Find the minimum length of fencing required, and the value of $x$ for which this minimum occurs.

Sketch a graph of fencing length against $x$ to confirm your answer.

The profit, $\$y$, generated from the sale of $x$ laptops is given by the formula $y=600x+15x^2-x^3.$ Find the number of laptops, $x$, which maximizes profit, and determine the maximum profit.

An open-top water tank is in the shape of a cuboid. The tank has a square base of side length $s \mathrm{m}$ and has a volume of $216{ \mathrm{m}}^3$.

Find expression in terms of $s,$ for the height of the tank.

An open-top water tank is in the shape of a cuboid. The tank has a square base of side length $s \mathrm{m}$ and has a volume of $216{ \mathrm{m}}^3$.

Find expression in terms of $s,$ for the surface area of the tank.

An open-top water tank is in the shape of a cuboid. The tank has a square base of side length $s \mathrm{m}$ and has a volume of $216{ \mathrm{m}}^3$.

Given that the surface area of the open-top water tank is minimized. Find the value of $s$.

A $150 \mathrm{m}$ length of wire is cut into two pieces. One of the pieces is bent to form a square with side length $s$, and the other piece is bent to form a rectangle with a side length that is twice its width. Find the value of $s$ which minimizes the sum of the areas enclosed by the square and rectangle.

[Enter the value excluding units]

An architect wants to design a rectangular water tank with an open top that can hold a capacity of $10 {\mathrm{m}}^3$. The materials needed for the tank cost $\$10$ per square metre for the base, and $\$6$ per square metre for the sides. The length of the base must be twice its width. Find the minimum cost of the material required to build the container. Give your answer to the nearest dollar.

[Enter the value excluding units]