Asked by Hi
The figure below shows a solid cone if the total surface area of the cone is 1496 CM determine the slanting height of the cone if the diameter is 28m
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Answered by
GPT-4o mini
To find the slant height of the cone given the total surface area and the diameter, we can use the formula for the total surface area of a cone:
\[
\text{Total Surface Area} = \pi r (r + l)
\]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
### Step 1: Convert diameter to radius
Given the diameter of the cone is 28 m, we can find the radius:
\[
r = \frac{\text{diameter}}{2} = \frac{28}{2} = 14 \text{ m}
\]
### Step 2: Substitute the known values into the total surface area formula
The total surface area is given as 1496 cm². However, since we have calculated the radius in meters, we need to convert the surface area to square meters.
Since \( 1 \text{ cm}^2 = 0.0001 \text{ m}^2 \):
\[
1496 \text{ cm}^2 = 1496 \times 0.0001 \text{ m}^2 = 0.1496 \text{ m}^2
\]
Now substitute \( r \) and the surface area into the formula:
\[
0.1496 = \pi (14) (14 + l)
\]
### Step 3: Solve for \( l \)
First, calculate the area using the value of \( \pi \) (approximately 3.14):
\[
0.1496 = 3.14 \cdot 14 \cdot (14 + l)
\]
Calculate \( 3.14 \cdot 14 \):
\[
3.14 \cdot 14 = 43.96
\]
Then, substitute this back:
\[
0.1496 = 43.96 (14 + l)
\]
### Step 4: Isolate \( l \)
Now divide both sides by 43.96:
\[
\frac{0.1496}{43.96} = 14 + l
\]
Calculating the left side:
\[
0.0034 \approx 14 + l
\]
Now isolate \( l \) by subtracting 14 from both sides:
\[
l = 0.0034 - 14
\]
This will yield:
\[
l \approx -13.9966
\]
This result appears unreasonable as the slant height cannot be negative. It's possible that there has been a misunderstanding regarding the measurement units or values provided. Please double-check the dimensions of the cone or any given values. Normally, if everything was accurate, the mathematics should yield a valid, positive slant height.
If there’s anything else or if the values need revision, let me know!
\[
\text{Total Surface Area} = \pi r (r + l)
\]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
### Step 1: Convert diameter to radius
Given the diameter of the cone is 28 m, we can find the radius:
\[
r = \frac{\text{diameter}}{2} = \frac{28}{2} = 14 \text{ m}
\]
### Step 2: Substitute the known values into the total surface area formula
The total surface area is given as 1496 cm². However, since we have calculated the radius in meters, we need to convert the surface area to square meters.
Since \( 1 \text{ cm}^2 = 0.0001 \text{ m}^2 \):
\[
1496 \text{ cm}^2 = 1496 \times 0.0001 \text{ m}^2 = 0.1496 \text{ m}^2
\]
Now substitute \( r \) and the surface area into the formula:
\[
0.1496 = \pi (14) (14 + l)
\]
### Step 3: Solve for \( l \)
First, calculate the area using the value of \( \pi \) (approximately 3.14):
\[
0.1496 = 3.14 \cdot 14 \cdot (14 + l)
\]
Calculate \( 3.14 \cdot 14 \):
\[
3.14 \cdot 14 = 43.96
\]
Then, substitute this back:
\[
0.1496 = 43.96 (14 + l)
\]
### Step 4: Isolate \( l \)
Now divide both sides by 43.96:
\[
\frac{0.1496}{43.96} = 14 + l
\]
Calculating the left side:
\[
0.0034 \approx 14 + l
\]
Now isolate \( l \) by subtracting 14 from both sides:
\[
l = 0.0034 - 14
\]
This will yield:
\[
l \approx -13.9966
\]
This result appears unreasonable as the slant height cannot be negative. It's possible that there has been a misunderstanding regarding the measurement units or values provided. Please double-check the dimensions of the cone or any given values. Normally, if everything was accurate, the mathematics should yield a valid, positive slant height.
If there’s anything else or if the values need revision, let me know!
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