Question
Nora bought a jewelry box. She wants to paint all of the exterior faces of the jewelry box. How much paint does she need?
Solve on paper, then enter your answer on Zearn. You can use the calculator to help you solve.
Right trapezoidal prism with side lengths labeled 3 in, 13 in, 8 in, 5 in and 9 in.
Nora's jewelry box
Nora needs
square inches of paint.
Solve on paper, then enter your answer on Zearn. You can use the calculator to help you solve.
Right trapezoidal prism with side lengths labeled 3 in, 13 in, 8 in, 5 in and 9 in.
Nora's jewelry box
Nora needs
square inches of paint.
Answers
GPT-4o mini
To find out how much paint Nora needs for the exterior faces of her jewelry box, which is a right trapezoidal prism, we need to calculate the surface area of the box.
1. **Identify the dimensions:** The dimensions provided are:
- Base lengths of the trapezoid: 3 in (shorter base) and 13 in (longer base)
- Height of the trapezoid (perpendicular distance between the two bases): 8 in
- Length of the prism (thickness): 5 in
- Height of the prism (other side): 9 in (this doesn't directly affect the surface area calculation for a trapezoidal prism but is important to recognize).
2. **Calculate the area of the trapezoidal face:**
The area \(A\) of the trapezoid can be calculated using the formula:
\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\]
where \(b_1\) and \(b_2\) are the lengths of the two bases, and \(h\) is the height.
Substituting the values:
\[
A = \frac{1}{2} \times (3 \, \text{in} + 13 \, \text{in}) \times 8 \, \text{in} = \frac{1}{2} \times 16 \, \text{in} \times 8 \, \text{in} = 64 \, \text{square inches}
\]
3. **Calculate the area of the rectangular faces:**
The trapezoidal prism has the following rectangular faces:
- Two bases (the trapezoidal faces): \(2 \times A\)
- Two lateral rectangular faces along the length of the prism:
- One face with dimensions (3 in by 5 in): \(3 \, \text{in} \times 5 \, \text{in} = 15 \, \text{square inches}\)
- One face with dimensions (13 in by 5 in): \(13 \, \text{in} \times 5 \, \text{in} = 65 \, \text{square inches}\)
- The other two sides will be:
- One face with dimensions (9 in by 5 in): \(9 \, \text{in} \times 5 \, \text{in} = 45 \, \text{square inches}\)
- The opposite side will be the same as this: \(45 \, \text{square inches}\)
4. **Calculate total surface area:**
Combining all the areas:
- Areas of the two trapezoidal bases: \(2 \times 64 \, \text{sq in} = 128 \, \text{sq in}\)
- Areas of the rectangular sides:
- Longer side: \(15 \, \text{sq in} + 65 \, \text{sq in} + 45 \, \text{sq in} + 45 \, \text{sq in} = 170 \, \text{sq in}\)
So, the total surface area \(S\) is:
\[
S = 128 \, \text{sq in} + 170 \, \text{sq in} = 298 \, \text{square inches}
\]
Therefore, Nora needs **298 square inches** of paint.
1. **Identify the dimensions:** The dimensions provided are:
- Base lengths of the trapezoid: 3 in (shorter base) and 13 in (longer base)
- Height of the trapezoid (perpendicular distance between the two bases): 8 in
- Length of the prism (thickness): 5 in
- Height of the prism (other side): 9 in (this doesn't directly affect the surface area calculation for a trapezoidal prism but is important to recognize).
2. **Calculate the area of the trapezoidal face:**
The area \(A\) of the trapezoid can be calculated using the formula:
\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\]
where \(b_1\) and \(b_2\) are the lengths of the two bases, and \(h\) is the height.
Substituting the values:
\[
A = \frac{1}{2} \times (3 \, \text{in} + 13 \, \text{in}) \times 8 \, \text{in} = \frac{1}{2} \times 16 \, \text{in} \times 8 \, \text{in} = 64 \, \text{square inches}
\]
3. **Calculate the area of the rectangular faces:**
The trapezoidal prism has the following rectangular faces:
- Two bases (the trapezoidal faces): \(2 \times A\)
- Two lateral rectangular faces along the length of the prism:
- One face with dimensions (3 in by 5 in): \(3 \, \text{in} \times 5 \, \text{in} = 15 \, \text{square inches}\)
- One face with dimensions (13 in by 5 in): \(13 \, \text{in} \times 5 \, \text{in} = 65 \, \text{square inches}\)
- The other two sides will be:
- One face with dimensions (9 in by 5 in): \(9 \, \text{in} \times 5 \, \text{in} = 45 \, \text{square inches}\)
- The opposite side will be the same as this: \(45 \, \text{square inches}\)
4. **Calculate total surface area:**
Combining all the areas:
- Areas of the two trapezoidal bases: \(2 \times 64 \, \text{sq in} = 128 \, \text{sq in}\)
- Areas of the rectangular sides:
- Longer side: \(15 \, \text{sq in} + 65 \, \text{sq in} + 45 \, \text{sq in} + 45 \, \text{sq in} = 170 \, \text{sq in}\)
So, the total surface area \(S\) is:
\[
S = 128 \, \text{sq in} + 170 \, \text{sq in} = 298 \, \text{square inches}
\]
Therefore, Nora needs **298 square inches** of paint.