DP Mathematics: Applications and Interpretation Questionbank

Given that the total external surface area of the box is $1200 {\text{cm}}^2$, show that the volume of the box may be expressed as $V=300\sqrt{3}x-\frac{9}{4}{x}^3$.

Write down the value of the iron in the form $a\times {10}^k$ where $1\le a<10 , k\in \mathrm{\mathbb{Z}}$.

Calculate James’s estimate of its volume, in ${\text{km}}^3$.

The actual volume of the asteroid is found to be $6.074\times {10}^6 {\text{km}}^3$.

Find the percentage error in James’s estimate of the volume.

The total surface of the candy is coated in chocolate. It is known that $1$ gram of the chocolate covers an area of $240 {\text{mm}}^2$.

Calculate the weight of chocolate required to coat one piece of candy.

Calculate the total surface area of one piece of candy.

A storage container consists of a box of length $90 \text{cm}$, width $42 \text{cm}$ and height $34 \text{cm}$, and a lid in the shape of a half-cylinder, as shown in the diagram. The lid fits the top of the box exactly. The total exterior surface of the storage container is to be painted.

Find the area to be painted.

Find the coordinates of station $\text{M}$.

Find the distance between stations $\text{A}$ and $\text{B}$.

Write down the height of station $\text{M}$, in metres, above the ground.

Write down the perimeter of the base of the hat in terms of $\mathrm{\pi }$.

Find the value of $\theta$.

Find the surface area of the outside of the hat.

Find the expression $\frac{dV}{d\theta }$.

Solve algebraically $\frac{dV}{d\theta }=0$ to find the value of $\theta$ that will maximize the volume, $V$.

Show that $r=\frac{6}{2+\theta }$.

Find an expression for $V$ in terms of $\theta$.

Find the angle of depression from $\text{M}$ to $\text{N}$.

Find $\text{CV}$.

Hence or otherwise, show that the volume of the reservoir is $29 600 {\text{m}}^3$.

To avoid water leaking into the ground, the five interior sides of the reservoir have been painted with a watertight material.

Find the area that was painted.

By finding an appropriate value, determine whether Joshua is correct.

The front view of a doghouse is made up of a square with an isosceles triangle on top.

The doghouse is $1.35 \text{m}$ high and $0.9 \text{m}$ wide, and sits on a square base.

The top of the rectangular surfaces of the roof of the doghouse are to be painted.

Find the area to be painted.

Find $\text{FÂO}$, the angle the rope makes with the ground.

Find the length of the rope connecting $\text{A}$ to $\text{F}$.

Find $\overrightarrow{\text{AF}}$.

Find the length of the rope.

Find $\text{FÂO}$, the angle the rope makes with the platform.

Helen is building a cabin using cylindrical logs of length 2.4 m and radius 8.4 cm. A wedge is cut from one log and the cross-section of this log is illustrated in the following diagram.

Find the volume of this log.

Find the equation of $\text{(XZ)}$.

Determine the exact length of $\text{[YZ]}$.

Given that the exact length of $\text{[XY]}$ is $\sqrt{32}$, find the size of $\text{XŶZ}$ in degrees.

A car departs from a point due north of Hamilton. It travels due east at constant speed to a destination point due North of Gaussville. It passes through the Edison, Isaacopolis and Fermitown districts. The car spends $30\%$ of the travel time in the Isaacopolis district.

Find the distance between Gaussville and the car’s destination point.

Calculate the radius of the base of the cone which has been removed.

Find the slant height of the cone-shaped container.

Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.

Find the height, $h$, of this cylinder-shaped container.

The factory director wants to increase the volume of coconut water sold per container.

State whether or not they should replace the cone-shaped containers with cylinder‑shaped containers. Justify your conclusion.

Write down the height of the cylinder.

Find the total volume of the trash can.

Find the height of the cylinder, h , of the new trash can, in terms of r.

Show that the volume, V cm³ , of the new trash can is given by

$V=110\pi r^3$.

Using your graphic display calculator, find the value of r which maximizes the value of V.

The designer claims that the new trash can has a capacity that is at least 40% greater than the capacity of the original trash can.

State whether the designer’s claim is correct. Justify your answer.

Show that the volume of a cone shaped glass is $160\text{ c}{\text{m}}^3$, correct to 3 significant figures.

Calculate the radius, $r$, of a hemisphere shaped glass.

Find the cost of $100\text{ c}{\text{m}}^3$ of chocolate mousse.

Show that there is $20\text{ c}{\text{m}}^3$ of orange paste in each special dessert.

Find the total cost of the ingredients of one special dessert.

Find the value of $x$.

Calculate the volume of this pan.

Find the radius of the sphere in cm, correct to one decimal place.

Find the value of $a$.

Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.

Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.

In the context of this model, state what the value of 19 represents.

Calculate the area of triangle EAD.

Calculate the total volume of the barn.

Calculate the length of MN.

Calculate the length of AE.

Show that Farmer Brown is incorrect.

Calculate the total length of metal required for one support.

Write down a formula for $A$, the surface area to be coated.

Express this volume in $\text{c}{\text{m}}^3$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find $\frac{\text{d}A}{\text{d}r}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Write down the value of x.

Calculate the volume of the paperweight.

1 cm³ of glass has a mass of 2.56 grams.

Calculate the mass, in grams, of the paperweight.

Find the volume of the money box.

A second money box is in the shape of a sphere and has the same volume as the cylindrical money box.

Find the diameter of the second money box.

Write down an equation for the area, $A$, of the curved surface in terms of $r$.

Find $\frac{\text{d}A}{\text{d}r}$.

Find the value of $r$ when the area of the curved surface is maximized.

Calculate the volume of the balloon.

Calculate the radius of the balloon following this increase.

Find the slant height of the cone-shaped container.

Calculate the volume of oil drained from Yao’s motorbike.

Yao then pours all the oil from the cuboids into an empty cylindrical container. The height of the oil in the container is $30$ cm.

Find the internal radius, $r$, of the container.

Write down an expression for $V$, the volume (cm³) of the speaker, in terms of $r$, $l$ and $\mathrm{\pi }$.

Write down an equation for the surface area of the speaker in terms of $r$, $l$ and $\mathrm{\pi }$.

Given the design constraint that $l=\frac{150-2\mathrm{\pi }{r}^2}{\mathrm{\pi }r}$, show that $V=150r-\frac{2\mathrm{\pi }{r}^3}{3}$.

Find $\frac{\text{d}V}{\text{d}r}$.

Using your answer to part (d), show that $V$ is a maximum when $r$ is equal to $\sqrt{\frac{75}{\mathrm{\pi }}} \text{cm}$.

Find the length of the cylinder for which $V$ is a maximum.

Calculate the maximum value of $V$.

Use your answer to part (f) to identify the shape of the speaker with the best quality of sound.