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 \( 170 \pi \mathrm{~cm}^{2} \)

To find the circumference of the circle, we start with the area formula \( A = \pi r^2 \). Given that the area is \( 25 \pi \, \text{cm}^2 \), we can set up the equation: \[ \pi r^2 = 25 \pi \] Dividing both sides by \( \pi \), we get \( r^2 = 25 \). This gives us \( r = 5 \, \text{cm} \). Now, using the circumference formula \( C = 2 \pi r \), we have: \[ C = 2 \pi (5) = 10 \pi \, \text{cm} \] For part (b), the total surface area \( A \) of a cylinder is given by the formula \( A = 2\pi r^2 + 2\pi rh \). The first term \( 2\pi r^2 \) accounts for the two circular ends. For our circle with radius \( r = 5 \, \text{cm} \), we substitute into the equation: \[ A = 2\pi(5^2) + 2\pi(5)(h) = 2\pi(25) + 10\pi h = 50\pi + 10\pi h \] Setting this equal to the total surface area \( 170\pi \): \[ 50\pi + 10\pi h = 170\pi \] To find \( h \), we can first subtract \( 50\pi \) from both sides: \[ 10\pi h = 120\pi \] Now, dividing both sides by \( 10\pi \), we find: \[ h = 12 \, \text{cm} \] So the radius of the circle is \( 5 \, \text{cm} \) and the height of the cylinder is \( 12 \, \text{cm} \).