5.7 Optimisation questions for IB Maths AI

Show that the volume of the box, $V$ cm³, is given by $V=x(40-2x)(25-2x)$.

Write down the set of possible values of $x$ for which the model is valid.

Find the maximum volume of the box and the value of $x$ that gives it.

Find the area of lawn watered.

Find the perimeter of the watered region.

Write down the value of $\theta$ in degrees.

Inés now plans a new flower bed in the shape of a sector with radius $r$ m and angle $\theta$ radians. The area of the flower bed must be $90{\text{ m}}^2$.

Show that $\theta =\frac{180}{r^2}$.

The whole perimeter of the flower bed will be edged with metal strip. Show that the length of strip needed, $P$ m, is given by

Find $\frac{\mathrm{d}P}{\mathrm{d}r}$.

Hence find the value of $r$ that minimizes $P$, and the minimum length of strip needed.

Find the corresponding value of $\theta$.

The metal strip costs 14.50 euros (EUR) per metre. Find the minimum cost of the strip. Give your answer correct to two decimal places.

Use the trapezoidal rule with all the values in the table to estimate the distance between the two stops.

The tram company's engineer, Mireille, models the velocity by

Find the distance between the stops according to this model.

Find the percentage difference between your answer to part (a) and your answer to part (b)(i), taking the value in part (b)(i) as the true value.

Use differentiation to find the maximum velocity of the tram according to Mireille's model.

Find the maximum acceleration of the tram according to the model, and the time at which it occurs.

For passenger comfort, the magnitude of the acceleration of a tram must never exceed 1.3 m s⁻².

Show that Mireille's model satisfies this condition.

Find the time at which the tram has travelled half of the distance between the stops.

On a new route, two stops are 900 m apart. Mireille uses a model of the same form,

where $T$ seconds is the journey time and $c$ is a positive constant.

Show that $c={\displaystyle \frac{10800}{T^4}}$.

Show that the greatest magnitude of the acceleration during the journey is ${\displaystyle \frac{10800}{T^2}}$.

Hence find the shortest journey time on the new route that satisfies the comfort condition in part (e).

Find the maximum velocity of the tram on this journey.

State one reason why Mireille's model may not be suitable for the new route.

Find $P^{\mathrm{\prime }}(n)$.

Find the number of cakes the bakery should make each day to maximise its profit.

Find the maximum daily profit.

Find the length of the arc of the sector.

Show that the radius of the base of the cone is $13.4$ cm, correct to three significant figures.

Find the height of the cone.

Find the volume of the cup.

Find the area of the curved surface of the cup.

Kenji now considers other values of $\theta$, where $0<\theta <2\pi$.

Show that the volume of the cup, $V{\text{ cm}}^3$, is given by

Find the value of $\theta$ that gives the greatest volume, and this greatest volume.

Explain why $\theta$ must be less than $2\pi$.

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.