The diagram shows a prism of length 10cm - OCR - GCSE Maths - Question 4 - 2023 - Paper 5

To find the correct value of b, we start with the relationships given:

Let h be the height in cm, then the base b can be expressed as: $b = h + 2$

The area A of the triangle (base and height) can be calculated as: $A = \frac{1}{2} \times b \times h = \frac{1}{2} \times (h + 2) \times h$

The volume V of the prism is given as: $V = A \times \text{length} = \left( \frac{1}{2} \times (h + 2) \times h \right) \times 10$

We set this equal to the given volume of 240 cm³: $\left( \frac{1}{2} \times (h + 2) \times h \right) \times 10 = 240$

Simplifying, we find: $\frac{(h + 2) \times h}{2} = 24$

$(h + 2) \times h = 48$

This results in the quadratic equation: $h^2 + 2h - 48 = 0$

Using the quadratic formula: $h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where a = 1, b = 2, and c = -48, we can find: $h = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times -48}}{2 \times 1}$ $h = \frac{-2 \pm \sqrt{4 + 192}}{2}$ $h = \frac{-2 \pm 14}{2}$

Calculating the positive root: $h = 6 ext{ cm}$

Now substituting back to find b: $b = h + 2 = 6 + 2 = 8 ext{ cm}$