(a) The diagram shows a pyramid standing on a cuboid. The dimensions of the cuboid are 4m by 3m by 2m and the slant edge of the pyramid is 5m. Calculet the ...

Answer Details

(a) Volume of the solid (pyramid on a cuboid).

From the diagram the cuboid has base \(4\text{ m} \times 3\text{ m}\) and height \(2\text{ m}\); the pyramid stands on the top \(4\text{ m} \times 3\text{ m}\) face and has a slant edge (apex to base corner) of \(5\text{ m}\).

Volume of the cuboid:

\[V_{\text{cuboid}} = 4 \times 3 \times 2 = 24 \text{ m}^3\]

Height of the pyramid. The apex is above the centre of the rectangular base. The distance from the centre to a base corner is half the diagonal of the \(4 \times 3\) rectangle:

\[\text{diagonal} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5 \text{ m}, \qquad \text{half-diagonal} = \tfrac{5}{2} = 2.5 \text{ m}\]

Using the right triangle (height, half-diagonal, slant edge):

\[h = \sqrt{5^2 - 2.5^2} = \sqrt{25 - 6.25} = \sqrt{18.75} \approx 4.330 \text{ m}\]

Volume of the pyramid:

\[V_{\text{pyramid}} = \tfrac{1}{3} \times (4 \times 3) \times h = \tfrac{1}{3} \times 12 \times 4.330 = 17.32 \text{ m}^3\]

Total volume of the shape:

\[V = 24 + 17.32 = 41.32 \text{ m}^3\]

Total volume \(\approx 41.32\text{ m}^3\) (to 2 d.p.).

(b) Value of x in the A.P.

The 2nd, 3rd and 4th terms are \(x-2\), \(5\) and \(x+2\). In an A.P. the difference between successive terms is constant, so the middle term equals the average of its neighbours (equivalently, consecutive differences are equal):

\[5 - (x-2) = (x+2) - 5\]\[7 - x = x - 3\]\[10 = 2x \quad\Rightarrow\quad x = 5\]

x = 5. (Check: terms are \(3, 5, 7\), common difference \(2\).)