To provide assistance, let's work through each question step by step.
Question 1
To compare the volumes of the two similarly shaped wedge polyhedrons:
Assuming you're given or can calculate the volume of both figures:
- You would find the volumes of both shapes.
- If Vol_A is the volume of Figure A and Vol_B is the volume of Figure B, then you find the ratio of their volumes.
If Vol_A = 125 and Vol_B = 1, then the volume ratio is 125:1.
If Vol_A = 5 and Vol_B = 1, the ratio is 5:1.
True statement:
- If the volumes are correctly calculated and you find the ratio is indeed 125:1, then "The two figures are similar because the volumes have a ratio of 125 : 1."
- If the ratio is 5:1, then "The two figures are similar because the volumes have a ratio of 5 : 1."
- If the calculations don't yield either of those ratios, choose "The two figures are not similar because the volumes are not the same."
Question 2
To find the scale factor between two similar boxes with volumes of 648 in³ and 24 in³:
-
Calculate the ratio of the volumes:
Scale Factor = (Volume of larger box) / (Volume of smaller box) = 648 / 24 = 27. -
The square of the scale factor based on volume gives a ratio of dimensions: √27 = approximately 5.2, but we aren't looking for dimensions, just for the derived values.
However, the numbers provided don't include 27, suggesting a misunderstanding of what's meant by scale factor in terms of volume ratios for dimensions.
Correct Response: None of the provided options (2, 3, 12, ½, 1) are appropriate if we take ratios at face value, and it might instead need to be reduced or expressed in decimal form.
Question 3
To find the volume of a similar rectangular pyramid with a scale factor of 23:
-
Find the original volume of the pyramid given: \[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Base Area = \( \text{Length} \times \text{Width} = 24 \times 22 = 528 , \text{ft}^2 \) Volume = \( \frac{1}{3} \times 528 \times 28 = \frac{14784}{3} = 4928 , \text{ft}^3 \)
-
Applying the volume scale factor (scale factor cubed): \[ \text{New Volume} = 4928 \times (23)^3 = 4928 \times 12167 \text{ (approx)} = 600,000 + \text{approx.}\]
Assuming available choices, calculate correctly and match closest numbers for relevant selections.
Question 4
To determine the volume of a rectangular prism dilating by a scale factor of 4:
-
Initial volume: \[ \text{Volume} = \text{Width} \times \text{Length} \times \text{Height} = 3 \times 4 \times 2 = 24 , m^3 \]
-
New dimensions after dilation: Each dimension multiplies by the scale factor:
- Width: \( 3 \times 4 = 12 , m \)
- Length: \( 4 \times 4 = 16 , m \)
- Height: \( 2 \times 4 = 8 , m \)
\[ \text{New Volume} = 12 \times 16 \times 8 = 1536 , m^3 \]
Correct Response: 1,536 m³
Question 5
For a triangular prism with dimensions 7 m (width), 12 m (length), and 8 m (height) dilating by a scale factor of 3:
-
Volume calculation of the triangular prism: \[ \text{Volume} = \frac{1}{2} \times \text{Base} \times \text{Height} \times \text{Length} \] Assuming \( \text{Base} = 7 \) m, \( \text{Height}_{triangular}\ = 8 \) m, \[ \text{Volume} = \frac{1}{2} \times 7 \times 8 \times 12 = \frac{1}{2} \times 672 = 336 , m^3 \]
-
New Volume Calculation using Scale Factor: \[ \text{New Volume} = 336 \times (3)^3 = 336 \times 27 = 9072 , m^3 \]
Correct Response: 9,072 m³
Please let me know if more information or clarification is needed on any of these questions!
1 answer