L2 Calculus
Question Bank
A company produces and sells smartphone cases. The daily profit $\$P$ (in hundreds of dollars) depends on the number of cases produced, $n$ (in hundreds), and is modelled by
$P(n) = -2n^2 + 16n - 10$
Use calculus to find:
(a) The number of cases that should be produced daily to maximize profit.
(b) The maximum daily profit.
A farmer has 120 metres of fencing to make a rectangular paddock. One side of the paddock is against an existing hedge, so it doesn't need fencing. What are the dimensions of the paddock with the largest possible area?
A farmer is building two rectangular paddocks alongside a dense hedge, one for goats and the other for sheep, as shown in the diagram. She has only 480m of fencing available and wants this to enclose the largest possible area. What is the largest possible combined area enclosed by the two pens?
A bakery sells cakes. The profit, $\$P$, from selling cakes at a price of $\$c$ each is modelled by the function $P(c) = 25c(18 - c)$. Use calculus to find the cake price that gives the maximum possible profit.
A rectangle is drawn inside the parabola $y = 12 - x^2$ with its base on the x-axis and its top corners on the parabola. Find the dimensions of the rectangle with the maximum possible area.
A roofing company is making gutters from a flat rectangular sheet of metal that is $40 \text{ cm}$ wide. They will fold up the edges on each side by a height of $x \text{ cm}$ to form an open-top gutter with a rectangular cross-section, as shown in the diagram below.
Use calculus to find the value of $x$ that will maximize the cross-sectional area. State the maximum area the gutter can carry.
Aroha wants to build a new rabbit enclosure with $16 \text{m}$ of material for the three new sides, as shown in the plan below. The side against the house does not need any material.
The total height of the enclosure is $2 + x$. Use calculus to find the value of $x$ that will maximize the area of the enclosure and state the maximum area.
A box is to be made from a piece of cardboard that is 50 cm by 20 cm by cutting of the corners (as shown in the diagram) to make a net for the box. Determine the height of the box that will give a maximum volume.
An open-top box with a square base is to be made from 108 cm² of cardboard. Find the maximum possible volume of the box.
A trapezium sits inside the parabola so that the two bottom corners lie on the x-intercepts and the two top corners touch the parabola. Find the maximum possible area of the trapezium.
A window composite shape consisting of a semicircle on top of a rectangle. The outer frame (perimeter) of the window is $3$ m. Find the dimensions of the window that give the maximum area, and state the maximum area.
A rectangular container with a square base and a lid must have a volume of 64 m³. The material for the base costs \$5/m², the lid costs \$3/m², and the sides cost \$2/m². Find the dimensions that will minimise the total cost.
An open box container with a square base is made for production. The container must have a volume of 32 m³. Use calculus to find the minimum possible surface area of the container.
A rectangle is drawn inside the parabola $y = c - x^2$ so that two of its vertices lie on the x-axis, and two of its vertices lie on the parabola, as shown in the diagram.
Show that the area of the rectangle is maximised when the width of the rectangle is $2\sqrt{\frac{c}{3}}$.
A triangle is drawn inside a parabola so that the top right corner touches the parabola, and the bottom left corner is at the origin (as shown). The parabola has the equation $y=-(mx-1)^2+5$, where $m$ is a positive constant.
Show that the triangle with the maximum possible area occurs when the width of the triangle is $\frac{2}{m}$.
Find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum.
A cylindrical can with a lid must hold a volume of 500 cm³. Find the radius, $r$, that will minimise the surface area of the can.
