DETAILS MY NOTES AUFEXC4 7.5.010.MI. Find the surface area of the figure when \( d=4 \mathrm{~cm} \). For calculations involving \( \pi \), give both the exact value and an approximation to the nearest hundredth of a centimete \( \mathrm{cm}^{2} \approx \square \mathrm{~cm}^{2} \)

To find the surface area of the figure when \( d = 4 \, \text{cm} \), we need to know the shape of the figure. Since the problem does not specify the shape of the figure, we will assume it is a cylinder. The surface area of a cylinder is given by the formula: \[ A = 2\pi r^2 + 2\pi rh \] where: - \( r \) is the radius of the base of the cylinder - \( h \) is the height of the cylinder Given that \( d = 4 \, \text{cm} \), we can find the radius \( r \) and height \( h \) of the cylinder. The diameter \( d \) is related to the radius \( r \) by the formula: \[ d = 2r \] So, we can find the radius \( r \) by dividing the diameter by 2: \[ r = \frac{d}{2} = \frac{4}{2} = 2 \, \text{cm} \] Now, we need to find the height \( h \) of the cylinder. Since the problem does not provide the height, we will assume it is also 4 cm. Substitute the values of \( r \) and \( h \) into the surface area formula: \[ A = 2\pi (2)^2 + 2\pi (2)(4) \] Now, we can calculate the surface area of the cylinder using the exact value of \( \pi \) and then approximate it to the nearest hundredth of a centimeter squared. Calculate the value by following steps: - step0: Calculate: \(2\pi \times 2^{2}+2\pi \times 2\times 4\) - step1: Multiply: \(8\pi +2\pi \times 2\times 4\) - step2: Multiply the terms: \(8\pi +16\pi \) - step3: Collect like terms: \(\left(8+16\right)\pi \) - step4: Add the numbers: \(24\pi \) The exact surface area of the cylinder is \( 24\pi \, \text{cm}^2 \). Now, let's approximate this value to the nearest hundredth of a centimeter squared. Calculate the value by following steps: - step0: Calculate: \(24\pi \) The approximate surface area of the cylinder to the nearest hundredth of a centimeter squared is \( 75.40 \, \text{cm}^2 \).