Mathematics 4024 · O Level · Surface area and volume

Surface area and volume — practice question

The diagram depicts a solid cone with radius $r$ cm, height $h$ cm and slant height $l$ cm. Each cone’s slant height is $4$ cm greater than its radius. Use $\pi = 3$ throughout this question.
(a)[2]

Show that the total surface area $A\text{ cm}^2$ of each cone can be written as $A = 6r(r+2)$.

(b)[1]

Fill in the table for $A = 6r(r+2)$.

(c)[2]

On the grid opposite, plot the graph of $A = 6r(r+2)$.

(d)[2]

Find $h$ as an expression in terms of $r$.

(e)[2]

A cone has height $12\text{ cm}$. Calculate its radius.

(f(i))[1]

Another cone has a surface area of $200\text{ cm}^2$. Use your graph to find the radius of this cone.

(f(ii))[2]

This cone is placed in a box of height $p$ cm, where $p$ is an integer. Find the smallest possible value of $p$.

Worked solution & mark scheme

This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: $\pi r^2 + \pi r(r+4)$, which simplifies to $6\pi(r+2)$

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