26 A circle has an area of \( 25 \pi \mathrm{~cm}^{2} \). (a) Work out the circumference of the circle. Give your answer in terms of \( \pi \). (b) Two of the circles are used as the ends of a cylinder, with height \( h \mathrm{~cm} \). The total surface area of the cylinder is \( 170 \pi \mathrm{~cm}^{2} \). Work out the value of \( h \). [3] [3]

To find the circumference of the circle given its area, we start with the formula for area: \[ A = \pi r^2 \] Given that the area \( A = 25 \pi \) cm², we set up the equation: \[ \pi r^2 = 25 \pi \] Dividing both sides by \( \pi \): \[ r^2 = 25 \quad \Rightarrow \quad r = 5 \text{ cm} \] Now, using the formula for circumference \( C = 2\pi r \): \[ C = 2\pi \times 5 = 10\pi \text{ cm} \] Next, for the cylinder, we use the formula for total surface area: \[ \text{Total Surface Area} = 2\pi r^2 + 2\pi rh \] Given the total surface area is \( 170\pi \): \[ 2\pi (5^2) + 2\pi (5)h = 170\pi \] Calculating \( 2\pi (5^2) \): \[ 2\pi \times 25 = 50\pi \] Substituting this into the surface area equation: \[ 50\pi + 10\pi h = 170\pi \] Now, we simplify: \[ 10\pi h = 170\pi - 50\pi \] \[ 10\pi h = 120\pi \] Dividing both sides by \( 10\pi \): \[ h = 12 \text{ cm} \] Thus, the circumference of the circle is \( 10\pi \text{ cm} \) and the height of the cylinder is \( 12 \text{ cm}. \)