Let's solve each question step by step.
Question 1: Surface Areas of Spheres
The formula for the surface area (SA) of a sphere is: \[ \text{SA} = 4\pi r^2 \]
For sphere A (radius = 6 in): \[ \text{SA}_A = 4\pi (6^2) = 4\pi (36) = 144\pi , \text{in}^2 \]
For sphere B (radius = 24 in): \[ \text{SA}_B = 4\pi (24^2) = 4\pi (576) = 2304\pi , \text{in}^2 \]
Now, comparing the surface areas: To find how many times larger sphere B's surface area is than sphere A's: \[ \frac{\text{SA}_B}{\text{SA}_A} = \frac{2304\pi}{144\pi} = \frac{2304}{144} = 16 \]
Answer: Sphere A has a surface area of \(144\pi\) in² and sphere B has a surface area of \(2304\pi\) in². This means that sphere B’s surface area is 16 times as large as sphere A’s.
Question 2: Comparing Surface Areas of Rectangular Prisms
Surface area comparison: Given that:
- Surface area of prism A = 25 sq. units
- Surface area of prism B = 100 sq. units
Prism B's surface area is: \[ \frac{100}{25} = 4 \]
Since area scales with the square of the dimensions, if the surface area is 4 times greater, the dimensions must be: \[ \sqrt{4} = 2 \]
Answer: Since prism B’s surface area is 4 times as large as prism A’s surface area, prism B’s dimensions are twice as large as prism A’s.
Question 3: Surface Area of a Dilated Triangular Prism
If the triangular prism has a surface area of 486 in² and is dilated by a scale factor of 13, the new surface area \( A' \) is calculated as follows: \[ A' = \text{original area} \times \text{scale factor}^2 \] \[ A' = 486 \times 13^2 = 486 \times 169 = 82,014 , \text{in}^2 \]
However, this doesn't match the given options. Let's recalculate using the correct factor. Scaling the prism's dimensions by 13 means the surface area scales by \(13^2\): \[ A' = 486 \times 169 = 82,014 \]
None of the provided options is correct. Please double-check the scale factor for proper application.
Question 4: Surface Area of a Dilated Cylinder
The formula for the surface area of a cylinder is: \[ \text{SA} = 2\pi rh + 2\pi r^2 \]
For a cylinder with:
- Height \( h = 15 \) cm
- Radius \( r = 8 \) cm Calculating the original surface area: \[ \text{SA}_{\text{original}} = 2\pi(8)(15) + 2\pi(8^2) \] \[ = 240\pi + 128\pi = 368\pi , \text{cm}^2 \]
If the cylinder is dilated by a scale factor of 4: The new surface area will scale by \( 4^2 = 16 \): \[ \text{SA}_{\text{new}} = 368\pi \times 16 = 5,888\pi , \text{cm}^2 \]
Answer: \(5,888\pi , \text{cm}^{2}\)
Question 5: Original Surface Area after Dilation
If the dilated figure has a surface area of \( 10,990.8 , \text{in}^2 \) and was dilated by a scale factor of 6, the original surface area \( A \) can be found using: \[ A = \frac{\text{dilated area}}{\text{scale factor}^2} \] \[ A = \frac{10,990.8}{6^2} = \frac{10,990.8}{36} = 304.2 , \text{in}^2 \]
However, the responses provided don't match this calculation. Ensure to check the scale factor presented and correct application for proper results aggregation.
Let me know if you need further help!
1 answer