3.1 Volume and surface area in three dimensions questions for IB Maths AI

Calculate the volume of Mars, in km³. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Write down the volume of Mars in m³.

The mass of Mars is $6.42\times {10}^{23}$ kg.

Calculate the mean density of Mars, in kg m⁻³.

Calculate the volume of the tank, in m³.

Given that 1 m³ $=$ 1000 litres, write down the volume of the tank in litres. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

The town uses $1.35\times {10}^7$ litres of water each day. No water enters the tank.

Find the number of complete days that a full tank can supply the town.

The radius and the depth of the tank were each measured correct to the nearest metre.

Find the upper bound of the volume of the tank.

The lower bound of the volume of the tank is 94 685.6 m³, correct to one decimal place.

Find the largest possible percentage error in using your answer to part (a) as the volume of the tank.

During one storm, 12 mm of rain fell on the area that drains into the tank. This area is 3.6 km².

(i) Find the volume of rain, in m³, that fell on this area. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

(ii) Find this volume as a percentage of the volume of the tank found in part (a).

Show that $h={\displaystyle \frac{500}{\pi r^2}}-{\displaystyle \frac{2r}{3}}$.

Show that the total cost, $C$ USD, of building the silo is given by

Explain why the model is only valid for $0<r<6.20$.

Find ${\displaystyle \frac{\mathrm{d}C}{\mathrm{d}r}}$.

Find the value of $r$ and the value of $h$ that minimize the cost, and find this minimum cost to the nearest dollar.

Use the second derivative to justify that this value of $r$ gives a minimum cost.

A local planning regulation states that the total height of the silo, including the roof, must not exceed 9 m.

Show that the radius must satisfy ${\displaystyle \frac{500}{\pi r^2}}+{\displaystyle \frac{r}{3}}\le 9$.

Find the least value of $r$ that satisfies the regulation.

Explain why the cheapest silo that satisfies the regulation has the radius found in part (g)(ii), and find its cost to the nearest dollar.

Find the percentage increase in the minimum cost caused by the regulation.

Suggest one further cost that the cooperative should include in the model before deciding on the radius of the silo.

Write down the set of possible values of $x$ for this context.

Find the possible values of $x$ for which the volume of the tray is $1800{\text{ cm}}^3$.

The equation in part (b) has a third solution. Explain why it is not a possible value of $x$.

Calculate the volume of one red blood cell, in m³, according to Ingrid's model. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

An adult has 5.0 litres of blood, and 45% of the volume of the blood is made up of red blood cells. 1 litre $={10}^{-3}$ m³.

Estimate the number of red blood cells in the adult's body. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

A medical textbook states that the adult has $2.5\times {10}^{13}$ red blood cells.

Taking the textbook value as exact, calculate the percentage error in Ingrid's estimate.

Each red blood cell contains $2.7\times {10}^8$ haemoglobin molecules.

Using your answer to part (b), find the total number of haemoglobin molecules in the adult's red blood cells.

Oxygen is absorbed through the surface of the red blood cells.

Calculate the total surface area of one red blood cell according to Ingrid's model.

Real red blood cells are thinner in the middle than at the edge.

State whether Ingrid's model is likely to overestimate or underestimate the volume of one red blood cell.

Write down the value of $d$ and a system of three equations satisfied by $a$, $b$ and $c$.

Find the values of $a$, $b$ and $c$.

Use calculus to find the maximum internal radius of the decanter and the height at which it occurs.

Find the internal radius at the narrowest point of the neck.

Find the capacity of the decanter, in litres.

Before fitting the curve, Lucía measured the internal radius of a prototype every 4 cm.

Use the trapezoidal rule with the cross-sectional areas $\pi r^2$ at these heights to estimate the capacity of the prototype, in cm³.

Find the percentage difference between your answers to parts (e) and (f)(i), taking the answer to part (e) as the true value.

The decanter stands upright and 1 litre of wine is poured into it.

Find the depth of the wine.

Wine is poured into the empty decanter at a constant rate of $50{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate at which the depth of the wine is increasing when the depth is 10 cm.

Lucía also makes a larger version of the decanter in which every length is multiplied by 1.2.

Find the capacity of the larger decanter, in litres.

Give one reason why the capacity found in part (e) may be different from the volume of wine the decanter actually holds.