Question

A home gardener is building a greenhouse in the shape of a hemisphere. The radius of the hemisphere is 12.3 meters, and construction costs are $25 per 100 square meters. What is the total cost of construction?

a.$59.41
b.$237.65
c.$475.39
d.$950.58

Answer 1

The total cost of construction is approximately $237.72. Therefore, the correct answer is (b) $237.65.

To find the total cost of construction, we need to calculate the surface area of the hemisphere and then use the construction cost per square meter.

The surface area
`\( A \)` of a hemisphere is given by the formula:

`\[ A = 2 \pi r^2 \]`

where
`\( r \)` is the radius.

Given that the radius
`\( r \)` is 12.3 meters, we can substitute this value into the formula:

`\[ A = 2 \pi (12.3)^2 \]`

Now, we can calculate the surface area and then find the total cost:

`\[ \text{Total cost} = (A)/(100) * \text{Construction cost per 100 square meters} \]`

Let's perform the calculations:

`\[ A = 2 \pi (12.3)^2 \]`

`\[ A \approx 2 * 3.14159 * (12.3)^2 \]`

`\[ A \approx 2 * 3.14159 * 151.29 \]`

`\[ A \approx 2 * 475.43 \]`

`\[ A \approx 950.86 \]`

Now, the total cost:

`\[ \text{Total cost} = (950.86)/(100) * 25 \]`

`\[ \text{Total cost} \approx 9.5086 * 25 \]`

`\[ \text{Total cost} \approx 237.715 \]`

So, the total cost of construction is approximately $237.72. Therefore, the correct answer is (b) $237.65.

Answer 2

To find the cost of constructing a hemisphere-shaped greenhouse with a radius of 12.3 meters, first calculate the surface area of the hemisphere, then multiply that by the construction cost per unit area. The correct total construction cost is approximately $237.63.

Step-by-step explanation:

The cost of constructing a greenhouse in the shape of a hemisphere with a radius of 12.3 meters is calculated by finding the surface area of the hemisphere and then multiplying it by the construction cost per unit area. The surface area (A) of a hemisphere is given by the formula A = 2πr², where r is the radius of the hemisphere. Using 12.3 meters for the radius, the surface area is:

A = 2π(12.3 m)² ≈ 2π(151.29 m²) ≈ 2π×151.29 m² ≈ 950.53 m².

Since the construction cost is $25 per 100 square meters, the total cost (C) can be computed as:

C = (950.53 m²) / (100 m²/$25) ≈ (950.53 / 100) × $25 ≈ $237.63.

Hence, the total construction cost for the hemisphere greenhouse is approximately $237.63, which matches option b.