Question
An old sailing ship has three triangular sails, Sail A, Sail B, and Sail C. Sail A, the smallest of the sails, measures 6 m along the base and between 5 and 8 m in height. Sail C, the largest of the sails, has an area less than twice the area of Sail A.
What could be the areas of the three sails? Complete the table by giving a possible base, height, and area for all three sails.
Comparison of Sail Dimensions and Areas
Base length
Height
Area
Sail A
Sail B
Sail C
The ship also has three circular portholes, Porthole A, Porthole B, and Porthole C. Porthole A, the smallest of the three portholes, has a circumference of between 2π and 6π feet. Porthole C, the largest of the portholes, has a circumference less than three times the circumference of Porthole A.
What could be the circumferences and areas of the three portholes? Complete the table by giving a possible radius, circumference, and area for all three portholes.
Comparison of Porthole Dimensions
Radius
Circumference
Area
Porthole A
Porthole B
Porthole C
What could be the areas of the three sails? Complete the table by giving a possible base, height, and area for all three sails.
Comparison of Sail Dimensions and Areas
Base length
Height
Area
Sail A
Sail B
Sail C
The ship also has three circular portholes, Porthole A, Porthole B, and Porthole C. Porthole A, the smallest of the three portholes, has a circumference of between 2π and 6π feet. Porthole C, the largest of the portholes, has a circumference less than three times the circumference of Porthole A.
What could be the circumferences and areas of the three portholes? Complete the table by giving a possible radius, circumference, and area for all three portholes.
Comparison of Porthole Dimensions
Radius
Circumference
Area
Porthole A
Porthole B
Porthole C
Answers
GPT-4o mini
Let's calculate the areas of the sails first:
### Sails
**Sail A:**
- Base: 6 m
- Height: Let's choose 7 m (within the given range of 5 to 8 m).
- Area = (Base * Height) / 2 = (6 * 7) / 2 = 21 m²
**Sail B:**
- To be moderately larger than Sail A, we can choose a base of 8 m and a height of 7 m.
- Area = (Base * Height) / 2 = (8 * 7) / 2 = 28 m²
**Sail C:**
- Sail C has an area less than twice the area of Sail A, so its maximum area can be calculated as less than 42 m² (2 * 21).
- Let's choose an area of 35 m² for Sail C.
- We assume a base of 10 m, so:
- Area = 35 m² ⇒ 35 = (Base * Height) / 2 ⇒ Base * Height = 70
- If we choose a base of 10 m, then Height = 70 / 10 = 7 m.
Now we can fill in the table for the sails:
| Dimension | Sail A | Sail B | Sail C |
|----------------|---------|---------|---------|
| Base length | 6 m | 8 m | 10 m |
| Height | 7 m | 7 m | 7 m |
| Area | 21 m² | 28 m² | 35 m² |
---
### Portholes
**Porthole A:**
- The circumference is between \(2\pi\) and \(6\pi\). Let's select \(4\pi\).
- To find the radius: Circumference = \(2\pi r\) ⇒ \(r = \frac{Circumference}{2\pi} = \frac{4\pi}{2\pi} = 2\) feet.
- Area = \(\pi r^2 = \pi (2^2) = 4\pi\) square feet.
**Porthole B:**
- Let's select \(5\pi\) for its circumference.
- Radius = \(r = \frac{5\pi}{2\pi} = 2.5\) feet.
- Area = \(\pi (2.5^2) = 6.25\pi\) square feet.
**Porthole C:**
- Given that its circumference is less than three times that of Porthole A, we can select \(8\pi\).
- Radius = \(r = \frac{8\pi}{2\pi} = 4\) feet.
- Area = \(\pi (4^2) = 16\pi\) square feet.
Now we can fill in the table for the portholes:
| Dimension | Porthole A | Porthole B | Porthole C |
|----------------|------------|------------|------------|
| Radius | 2 ft | 2.5 ft | 4 ft |
| Circumference | \(4\pi\) | \(5\pi\) | \(8\pi\) |
| Area | \(4\pi\) | \(6.25\pi\)| \(16\pi\) |
These tables summarize the possible areas and dimensions for the sails and portholes based on the constraints provided.
### Sails
**Sail A:**
- Base: 6 m
- Height: Let's choose 7 m (within the given range of 5 to 8 m).
- Area = (Base * Height) / 2 = (6 * 7) / 2 = 21 m²
**Sail B:**
- To be moderately larger than Sail A, we can choose a base of 8 m and a height of 7 m.
- Area = (Base * Height) / 2 = (8 * 7) / 2 = 28 m²
**Sail C:**
- Sail C has an area less than twice the area of Sail A, so its maximum area can be calculated as less than 42 m² (2 * 21).
- Let's choose an area of 35 m² for Sail C.
- We assume a base of 10 m, so:
- Area = 35 m² ⇒ 35 = (Base * Height) / 2 ⇒ Base * Height = 70
- If we choose a base of 10 m, then Height = 70 / 10 = 7 m.
Now we can fill in the table for the sails:
| Dimension | Sail A | Sail B | Sail C |
|----------------|---------|---------|---------|
| Base length | 6 m | 8 m | 10 m |
| Height | 7 m | 7 m | 7 m |
| Area | 21 m² | 28 m² | 35 m² |
---
### Portholes
**Porthole A:**
- The circumference is between \(2\pi\) and \(6\pi\). Let's select \(4\pi\).
- To find the radius: Circumference = \(2\pi r\) ⇒ \(r = \frac{Circumference}{2\pi} = \frac{4\pi}{2\pi} = 2\) feet.
- Area = \(\pi r^2 = \pi (2^2) = 4\pi\) square feet.
**Porthole B:**
- Let's select \(5\pi\) for its circumference.
- Radius = \(r = \frac{5\pi}{2\pi} = 2.5\) feet.
- Area = \(\pi (2.5^2) = 6.25\pi\) square feet.
**Porthole C:**
- Given that its circumference is less than three times that of Porthole A, we can select \(8\pi\).
- Radius = \(r = \frac{8\pi}{2\pi} = 4\) feet.
- Area = \(\pi (4^2) = 16\pi\) square feet.
Now we can fill in the table for the portholes:
| Dimension | Porthole A | Porthole B | Porthole C |
|----------------|------------|------------|------------|
| Radius | 2 ft | 2.5 ft | 4 ft |
| Circumference | \(4\pi\) | \(5\pi\) | \(8\pi\) |
| Area | \(4\pi\) | \(6.25\pi\)| \(16\pi\) |
These tables summarize the possible areas and dimensions for the sails and portholes based on the constraints provided.