A cuboid has dimensions $x\,\text{cm}$, $(5 - x)\,\text{cm}$ and $15\,\text{cm}$. (a) Show that the equation for the volume of the cuboid, $y\,\text{cm}^3$, is $y = 75x - 15x^2$. (b) Complete the table of values for $y = 75x - 15x^2$. $x$ 1 1.5 2 2.5 3 3.5 4 --- --- --- --- --- --- --- --- $y$ 60 78.75 90 90 78.75 60 (c) Draw the graph of $y = 75x - 15x^2$ for $1 \leq x \leq 4$. (d) [Volume of pyramid = $\frac{1}{3} \times \text{base area} \times \text{height}$] The diagram…

Mathematics (Syllabus D) 4024/21 — May/June 2023

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Topics Algebra and Graphs · Number · Geometry · Mensuration · Statistics · Probability · +3 more

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Fig. 1

A cuboid has dimensions $x\,\text{cm}$ , $(5 - x)\,\text{cm}$ and $15\,\text{cm}$ .

(a)

Show that the equation for the volume of the cuboid, $y\,\text{cm}^3$ , is $y = 75x - 15x^2$ .

(b)

Complete the table of values for $y = 75x - 15x^2$ .

$x$	1	1.5	2	2.5	3	3.5	4
$y$	60	78.75	90		90	78.75	60

(c)

Draw the graph of $y = 75x - 15x^2$ for $1 \leq x \leq 4$ .

Fig. 2

(d)

[Volume of pyramid = $\frac{1}{3} \times \text{base area} \times \text{height}$ ]

Fig. 3

The diagram shows a pyramid with a square base of side length $9\,\text{cm}$ .
The pyramid has height $x\,\text{cm}$ and volume $y\,\text{cm}^3$ .

(i)

Show that the equation for the volume of the pyramid is $y = 27x$ .

(ii)

By drawing a suitable straight line on the grid on page 2, find the height of the pyramid when the pyramid and the cuboid have the same volume.

______ $\text{cm}$

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