Question
At the Olympics, the winners of the gold, silver, and bronze medals stand on a tiered pedestal.
As labeled in the diagram, the height of pedestal 3 is 7 inches, the height of pedestal 2 is 11 inches, and the height of pedestal 1 is 15 inches. Assuming each pedestal is of equal width and depth, find the total surface area of the stand.
As labeled in the diagram, the height of pedestal 3 is 7 inches, the height of pedestal 2 is 11 inches, and the height of pedestal 1 is 15 inches. Assuming each pedestal is of equal width and depth, find the total surface area of the stand.
Answers
GPT-4o mini
To find the total surface area of the tiered pedestal stand that holds the gold, silver, and bronze medals, we need to calculate the surface area of all parts of the pedestal arrangement including the pedestal's top surfaces and the sides.
Let's start by defining the height and dimensions of each pedestal:
1. **Dimensions of each pedestal**:
- Pedestal 1 (15 inches height)
- Pedestal 2 (11 inches height)
- Pedestal 3 (7 inches height)
For simplicity, assume each pedestal has a width and depth of \( w \) and \( d \) respectively (let's assume \( w = d = 1 \) inch to include the total surface area calculation).
**Calculating the top surface areas**:
- **Top surface area of each pedestal**:
\[
\text{Area}_{\text{top}} = w \times d = 1 \times 1 = 1 \text{ square inch}
\]
- Since there are three pedestals, the total area of the tops:
\[
\text{Total Top Area} = 1 + 1 + 1 = 3 \text{ square inches}
\]
**Calculating the lateral surfaces**:
To find the lateral surface area of the pedestals, we calculate the area of the four sides for each pedestal.
1. **Lateral area of Pedestal 1 (15 inches height)**:
\[
\text{Lateral Area}_{1} = 4 \times (w \times \text{height}) = 4 \times (1 \times 15) = 60 \text{ square inches}
\]
2. **Lateral area of Pedestal 2 (11 inches height)**:
\[
\text{Lateral Area}_{2} = 4 \times (w \times \text{height}) = 4 \times (1 \times 11) = 44 \text{ square inches}
\]
3. **Lateral area of Pedestal 3 (7 inches height)**:
\[
\text{Lateral Area}_{3} = 4 \times (w \times \text{height}) = 4 \times (1 \times 7) = 28 \text{ square inches}
\]
**Total lateral surface area**:
Adding these areas together gives us:
\[
\text{Total Lateral Area} = 60 + 44 + 28 = 132 \text{ square inches}
\]
**Calculating the total surface area**:
The total surface area of the stand is the sum of the total top area and the total lateral area:
\[
\text{Total Surface Area} = \text{Total Top Area} + \text{Total Lateral Area} = 3 + 132 = 135 \text{ square inches}
\]
Thus, the total surface area of the stand is:
\[
\boxed{135} \text{ square inches}
\]
Let's start by defining the height and dimensions of each pedestal:
1. **Dimensions of each pedestal**:
- Pedestal 1 (15 inches height)
- Pedestal 2 (11 inches height)
- Pedestal 3 (7 inches height)
For simplicity, assume each pedestal has a width and depth of \( w \) and \( d \) respectively (let's assume \( w = d = 1 \) inch to include the total surface area calculation).
**Calculating the top surface areas**:
- **Top surface area of each pedestal**:
\[
\text{Area}_{\text{top}} = w \times d = 1 \times 1 = 1 \text{ square inch}
\]
- Since there are three pedestals, the total area of the tops:
\[
\text{Total Top Area} = 1 + 1 + 1 = 3 \text{ square inches}
\]
**Calculating the lateral surfaces**:
To find the lateral surface area of the pedestals, we calculate the area of the four sides for each pedestal.
1. **Lateral area of Pedestal 1 (15 inches height)**:
\[
\text{Lateral Area}_{1} = 4 \times (w \times \text{height}) = 4 \times (1 \times 15) = 60 \text{ square inches}
\]
2. **Lateral area of Pedestal 2 (11 inches height)**:
\[
\text{Lateral Area}_{2} = 4 \times (w \times \text{height}) = 4 \times (1 \times 11) = 44 \text{ square inches}
\]
3. **Lateral area of Pedestal 3 (7 inches height)**:
\[
\text{Lateral Area}_{3} = 4 \times (w \times \text{height}) = 4 \times (1 \times 7) = 28 \text{ square inches}
\]
**Total lateral surface area**:
Adding these areas together gives us:
\[
\text{Total Lateral Area} = 60 + 44 + 28 = 132 \text{ square inches}
\]
**Calculating the total surface area**:
The total surface area of the stand is the sum of the total top area and the total lateral area:
\[
\text{Total Surface Area} = \text{Total Top Area} + \text{Total Lateral Area} = 3 + 132 = 135 \text{ square inches}
\]
Thus, the total surface area of the stand is:
\[
\boxed{135} \text{ square inches}
\]