Let's go through the questions one by one:
Question 16
Original Rectangle Area
- Length = 9 units
- Width = 5 units
- Area = Length × Width = 9 × 5 = 45 square units
Dilated Rectangle Area
- Scale factor = 3
- New Length = 9 × 3 = 27 units
- New Width = 5 × 3 = 15 units
- New Area = 27 × 15 = 405 square units
Comparison of Areas
- Ratio of areas = 405 / 45 = 9
Response: 9 times greater
Question 17
Original Square Area
- Area = 16 square units
Dilated Square Area
- Scale factor = 0.5
- New area (scaled) = (0.5)^2 × Original area = 0.25 × 16 = 4 square units
Response: 4 square units
Question 18
Surface Area of Sphere A
- Radius = 5 in
- Surface Area = 4πr² = 4π(5)² = 4π(25) = 100π in²
Surface Area of Sphere B
- Radius = 10 in
- Surface Area = 4π(10)² = 4π(100) = 400π in²
Comparison of Surface Areas
- Ratio of surface areas = 400π / 100π = 4
Response: Sphere A has a surface area of 100π in² and sphere B has a surface area of 400π in², meaning sphere B’s surface area is 4 times as large as sphere A’s.
Question 19
Original Surface Area
The formula for the surface area of a rectangular prism is: \[ \text{Surface Area} = 2lw + 2lh + 2wh \]
Given:
- Length (l) = 4.4 m
- Height (h) = 10.2 m
- Width (w) = 3.6 m
Substituting values:
- 2(4.4)(10.2) + 2(4.4)(3.6) + 2(10.2)(3.6)
- 2(44.88) + 2(15.84) + 2(36.72)
- 89.76 + 31.68 + 73.44 = 195.88 m²
Dilated Surface Area
- Scale factor = 12
- New surface area = Original surface area × (scale factor)²
- New surface area = 195.88 × 144 = 28127.52 cm²
Response: 1,536 cm² does not match my calculations. Please double-check dimensions and units as they might have been mischaracterized.
Question 20
Original Volume
- Volume of a rectangular prism = Length × Width × Height = 2.5 ft × 2.5 ft × 4 ft = 25 ft³
Dilated Volume
- Scale factor = 3
- New Volume = (scale factor)³ × Original Volume = 3³ × 25 = 27 × 25 = 675 ft³
Response: 675 ft³
Question 21
Original Pyramid Volume
- Volume = (1/3) × base area × height
- Base Area = length × width = 8 in × 8 in = 64 in²
- Volume = (1/3) × 64 × 6 = 128 in³
Dilated Volume
- Scale factor = 4
- New Volume = (scale factor)³ × Original Volume = 4³ × 128 = 64 × 128 = 8192 in³
Response: 8,192 inches³
Question 22
This question requires an image to accurately apply the Pythagorean theorem and similar triangles to find OV. Please provide the specifics from the scenario to continue with this.
Question 23
To find the distance perpendicular to the hypotenuse, we can use the geometric mean: Let one segment of the hypotenuse be \( a = 32 \) inches and the other segment be \( b = 46 \) inches.
Using the geometric mean: \[ h = \sqrt{a \cdot b} \]
Calculating: \[ h = \sqrt{32 \times 46} \approx 38.4 \text{ inches} \]
Response: 38.4 inches
If you need any further calculations or assistance, feel free to ask!
1 answer