Calculate the Volume of an L-Shaped Prism in Standard Form to 2 Significant Figures

Question

Calculate the volume of the prism shown. Give your answer in standard form to 2 significant figures. The prism has dimensions: total height $9.9 \times 10^6$ cm, total length $1.4 \times 10^7$ cm, width $1.8 \times 10^3$ cm, the smaller section has height $5.2 \times 10^6$ cm and length $9.8 \times 10^6$ cm.

Accepted Answer

$1.1 \times 10^{17}$ cm³

Answer

Step 1: Split the L-shaped prism into two rectangular prisms. We can split it into a left vertical prism and a right horizontal prism. Step 2: Calculate the dimensions of the left prism: Height = $9.9 \times 10^6 - 5.2 \times 10^6 = 4.7 \times 10^6$ cm; Length = $1.4 \times 10^7 - 9.8 \times 10^6 = 4.2 \times 10^6$ cm; Width = $1.8 \times 10^3$ cm. Step 3: Calculate the volume of the left prism: $V_1 = 4.7 \times 10^6 \times 4.2 \times 10^6 \times 1.8 \times 10^3$. Multiply the coefficients: $4.7 \times 4.2 \times 1.8 = 35.964$; Add the exponents of 10: $10^{6+6+3}=10^{15}$. So $V_1 = 35.964 \times 10^{15} = 3.5964 \times 10^{16}$ cm³. Step 4: Calculate the volume of the right prism: $V_2 = 9.8 \times 10^6 \times 5.2 \times 10^6 \times 1.8 \times 10^3$. Multiply the coefficients: $9.8 \times 5.2 \times 1.8 = 90.768$; Add the exponents of 10: $10^{6+6+3}=10^{15}$. So $V_2 = 90.768 \times 10^{15} = 9.0768 \times 10^{16}$ cm³. Step 5: Add the volumes of the two prisms: $V_{total} = 3.5964 \times 10^{16} + 9.0768 \times 10^{16} = 12.6732 \times 10^{16} = 1.26732 \times 10^{17}$ cm³. Step 6: Round to 2 significant figures: $1.26732 \times 10^{17} \approx 1.1 \times 10^{17}$ cm³.