A barrel of oil has a height of 3 m and a diameter of 0.9 m . a) Estimate the volume of the oil in the barrel in metric units by rounding the diameter to the nearest meter and using 3 for \( \pi \). (Use \( V=\pi r^{2} h \).) b) Compute the actual volume of oil in the barrel. (Use \( V=\pi r^{2} h \).) a) The estimated volume of oil in the barrel is \( \square \mathrm{m}^{3} \). (Type an integer or a decimal.)

To estimate the volume of the oil in the barrel, we first round the diameter of 0.9 m to 1 meter. The radius \( r \) will then be \( \frac{1}{2} = 0.5 \) meters. Using the formula for the volume \( V = \pi r^{2} h \) with \( \pi \approx 3 \) and height \( h = 3 \) m: \[ V \approx 3 \times (0.5)^{2} \times 3 = 3 \times 0.25 \times 3 = 2.25 \, \text{m}^3 \] So, the estimated volume of oil in the barrel is approximately \( 2.25 \, \text{m}^3 \). For the actual volume, we use \( \pi \approx 3.14159 \) and the actual radius \( r = \frac{0.9}{2} = 0.45 \) m. Thus, \[ V = \pi \times (0.45)^{2} \times 3 \approx 3.14159 \times 0.2025 \times 3 \approx 1.899 \, \text{m}^3 \] So, the actual volume of oil in the barrel is approximately \( 1.899 \, \text{m}^3 \) (or you might round it to \( 1.90 \, \text{m}^3 \)).