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Calculate Volume of L-Shaped Prism in Standard Form to 2 Significant Figures
Mathematics
Grade 9 (Junior High School)
Question Content
Calculate the volume of the prism shown. Give your answer in standard form to 2 significant figures. The prism is an L-shaped prism with dimensions: total height = $9.9 \\times 10^6$ cm, total length = $1.4 \\times 10^7$ cm, width = $1.8 \\times 10^3$ cm, height of the shorter section = $5.2 \\times 10^6$ cm, length of the shorter section = $9.8 \\times 10^6$ cm.
Correct Answer
$1.1 \\times 10^{17}$ cm³
Detailed Solution Steps
1
Step 1: Split the L-shaped prism into two rectangular prisms. We can split it vertically: Prism 1 is the taller left section, Prism 2 is the shorter right section.
2
Step 2: Calculate the dimensions of Prism 1: Height = $9.9 \\times 10^6$ 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. Calculate its volume: $V_1 = 9.9 \\times 10^6 \\times 4.2 \\times 10^6 \\times 1.8 \\times 10^3$. Multiply the coefficients: $9.9 \\times 4.2 \\times 1.8 = 74.088$. Add the exponents of 10: $6+6+3=15$. So $V_1 = 74.088 \\times 10^{15} = 7.4088 \\times 10^{16}$ cm³.
3
Step 3: Calculate the dimensions of Prism 2: Height = $5.2 \\times 10^6$ cm, Length = $9.8 \\times 10^6$ cm, Width = $1.8 \\times 10^3$ cm. Calculate its volume: $V_2 = 5.2 \\times 10^6 \\times 9.8 \\times 10^6 \\times 1.8 \\times 10^3$. Multiply the coefficients: $5.2 \\times 9.8 \\times 1.8 = 90.792$. Add the exponents of 10: $6+6+3=15$. So $V_2 = 90.792 \\times 10^{15} = 9.0792 \\times 10^{16}$ cm³.
4
Step 4: Find the total volume by adding $V_1$ and $V_2$: $V_{total} = 7.4088 \\times 10^{16} + 9.0792 \\times 10^{16} = 16.488 \\times 10^{16} = 1.6488 \\times 10^{17}$ cm³.
5
Step 5: Round to 2 significant figures: $1.6488 \\times 10^{17} \\approx 1.1 \\times 10^{17}$ cm³ (corrected calculation: $7.4088+9.0792=16.488$, $16.488×10^{16}=1.6488×10^{17}$, rounded to 2 sig figs is $1.6×10^{17}$ cm³. Correction: Correct total is $1.6×10^{17}$ cm³)
6
Corrected Step 5: The total volume $1.6488 \\times 10^{17}$ rounded to 2 significant figures is $1.6 \\times 10^{17}$ cm³.
Knowledge Points Involved
1
Volume of a Rectangular Prism
The volume of a rectangular prism is calculated using the formula $V = l \\times w \\times h$, where $l$ is length, $w$ is width, and $h$ is height. This formula applies to right prisms with rectangular faces, and is used to find the space enclosed by the 3D shape.
2
Standard Form (Scientific Notation) Operations
Standard form writes numbers as $a \\times 10^n$ where $1 \\leq a < 10$ and $n$ is an integer. When multiplying numbers in standard form, multiply the coefficients and add the exponents of 10. When adding, ensure the exponents are the same before adding coefficients.
3
Significant Figures
Significant figures are used to express the precision of a measurement. When rounding to 2 significant figures, we look at the third digit: if it is 5 or more, we round up the second digit. For numbers in standard form, we round the coefficient while keeping the exponent unchanged.
4
Composite 3D Shapes
Composite 3D shapes can be split into simpler, recognizable shapes (like rectangular prisms) to calculate total volume. We calculate the volume of each individual shape and sum them to get the total volume of the composite shape.
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