The diagram shows a prism, ABCDEF. AB = 13 cm, AC = 20 cm, CF = 24 cm and angle ABC = 90°. (a) Calculate the total surface area of the prism. (b) Calculate the angle that AF makes with the base BCPE.

In the right-angled triangle ABC ($\angle ABC = 90^\circ$):

Given $AB = 13$ cm and $AC = 20$ cm.

The area of a right-angled triangle is $\frac{1}{2} \times \text{base} \times \text{height}$.

The total area of the two bases is $2 \times 98.7916 = 197.5832 \text{ cm}^2$.

The height of the prism is $CF = 24$ cm.
Area of face ABED:

Area of face BCFE:

Area of face ACFD:

Rounding to three significant figures:

The angle that line AF makes with the base BCPE (which is rectangle BCFE) is the angle between AF and its projection onto the base. Since AB is perpendicular to the base (as $\angle ABC = 90^\circ$ and it's a right prism), the projection of A onto the base is B. Therefore, the projection of AF onto the base is BF. The angle required is $\angle AFB$.

BF is the diagonal of the rectangle BCFE. In the right-angled triangle BCF:

We know $BC^2 = 231$ and $CF = 24$ cm.

Consider the right-angled triangle ABF, where $\angle ABF = 90^\circ$.
We have $AB = 13$ cm (opposite to $\angle AFB$) and $BF = \sqrt{807}$ cm (adjacent to $\angle AFB$).
Using the tangent function:

Rounding to one decimal place: