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Answer :
1. **Area of painted face:** 104 in²
2. **Surface area of pyramid:** 294.4 in²
3. **Width needed for volume:** 6 feet
4. **Volume of rectangular prism:** 3510 ft³
5. **Perimeter of rectangle:** 20 units
6. **Fourth vertex of rectangle:** (25, 3)
7. **Point at (-2, 0):** Point B
8. **Area of the figure:** 46 cm²
9. **Reflection axis:** y-axis, because the x-coordinate is the opposite
10. **Area of right triangle:** 350 m²
11. **Horizontal distance:** 25 units
12. **Vertical distance:** 24 units
let's solve each of these questions step by step:
1. **Area of the painted face of the brick:**
The vertices of the rectangle are given as (-4, 2), (-4, -11), (4, 2), and (4, -11).
The length is the distance between (-4, 2) and (4, 2) , which is:
[tex]\[ 4 - (-4) = 8 \text{ units} \][/tex]
The width is the distance between (-4, 2) and (-4, -11) , which is:
[tex]\[ 2 - (-11) = 13 \text{ units} \][/tex]
Therefore, the area of the rectangle is:
[tex]\[ 8 \times 13 = 104 \text{ square inches} \][/tex]
**Answer: 104 in²**
2. **Surface area of the pyramid:**
Base dimensions: [tex]\(6 \text{ in} \times 7 \text{ in}\)[/tex]
Larger triangular face height: [tex]\(8 \text{ in}\)[/tex]
Smaller triangular face height: [tex]\(8.2 \text{ in}\)[/tex]
To calculate the surface area, we consider both triangular faces and the rectangular base.
[tex]\[ \text{Base area} = 6 \times 7 = 42 \text{ in}^2 \][/tex]
**Answer: 147.2 in²**
3. **Width needed for the desired volume:**
Given:
[tex]\[ \text{Volume} = 93.6 \text{ ft}^3 \][/tex]
[tex]\[ \text{Length} = 7.8 \text{ ft}, \text{ Height} = 2 \text{ ft} \][/tex]
The formula for the volume of a rectangular prism is:
[tex]\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \][/tex]
[tex]\[ 93.6 = 7.8 \times \text{Width} \times 2 \][/tex]
[tex]\[ \text{Width} = \frac{93.6}{7.8 \times 2} = \frac{93.6}{15.6} = 6 \text{ ft} \][/tex]
**Answer: 6 feet**
4. **Volume of a rectangular prism:**
Given:
[tex]\[ \text{Length} = 22.5 \text{ ft}, \text{ Width} = 12 \text{ ft}, \text{ Height} = 13 \text{ ft} \][/tex]
The formula for volume:
[tex]\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \][/tex]
[tex]\[ \text{Volume} = 22.5 \times 12 \times 13 = 3510 \text{ ft}^3 \][/tex]
**Answer: 3510 ft³**
5. **Perimeter of the rectangle:**
The vertices of the rectangle are (-1, 3), (3, 3), (-1, -3), and (3, -3).
[tex]\[ 3 - (-1) = 4 \text{ units} \][/tex]
The width is the distance between (3, 3) and (3, -3), which is:
[tex]\[ 3 - (-3) = 6 \text{ units} \][/tex]
The perimeter is:
[tex]\[ 2 \times (4 + 6) = 20 \text{ units} \][/tex]
**Answer: 20 units**
6. **Fourth vertex needed to create a rectangle:**
Given vertices: (-15, 3) , (-15, -6) , (25, -6)
Therefore, the fourth vertex is:
[tex]\[ (25, 3) \][/tex]
**Answer: (25, 3)**
7. **Point located at (-2, 0):**
Given points:
- Point A: up one unit from the origin (0, 1)
- Point B: left two units from the origin (-2, 0)
- Point C: down two units from the origin (0, -2)
- Point D: right one unit from the origin (1, 0)
- Point E: right two units and up two units from the origin (2, 2)
- Point F: right two units and down two units from the origin (2, -2)
**Answer: Point B**
8. **Area of the figure:**
Given dimensions, it appears to be a trapezoid with:
- Top base: 9.2 cm
- Bottom base: 3 + 6.2 = 9.2 cm
- Height: 5 cm
The area of a trapezoid is given by:
[tex]\[ \text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height} \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times (9.2 + 9.2) \times 5 = \frac{1}{2} \times 18.4 \times 5 = 46 \text{ cm}^2 \][/tex]
**Answer: 46 cm²**
9. **Reflection over an axis:**
The point [tex]\((-8, -10)\)[/tex] is reflected to [tex]\( (8, -10)\).[/tex]
**Answer: y-axis, because the x-coordinate is the opposite**
10. **Area of a right triangle:**
[tex]\[ \text{Height} = 17.5 \text{ m}, \text{ Base} = 40 \text{ m} \][/tex]
The area of a right triangle is given by:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times 40 \times 17.5 = 350 \text{ m}^2 \][/tex]
**Answer: 350 m²**
11. **Horizontal distance from [tex]\((12, -2)\) to \((-13, -2)\)[/tex]:**
The distance is the difference in the x-coordinates, as the y-coordinates are the same:
[tex]\[ 12 - (-13) = 12 + 13 = 25 \text{ units} \][/tex]
**Answer: 25 units**
12. **Distance from [tex]\((-7, -14)\) to \((-7, 10)\)[/tex]:**
The distance is the difference in the y-coordinates, as the x-coordinates are the same:
[tex]\[ 10 - (-14) = 10 + 14 = 24 \text{ units} \][/tex]
**Answer: 24 units**
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