Sure, let's go through the entire problem step-by-step.
### Step 1: Determine the Base Radius of the Cone
We are given the area of the base of the cone, which is \( 152 \, \text{m}^2 \). The formula for the area of the base of a cone is:
[tex]\[ \pi r^2 \][/tex]
Given \( \pi r^2 = 152 \):
[tex]\[ r^2 = \frac{152}{\pi} \][/tex]
[tex]\[ r = \sqrt{\frac{152}{\pi}} \][/tex]
Evaluating the above expression, we find:
[tex]\[ r \approx 6.96 \, \text{m} \][/tex]
### Step 2: Determine the Height of the Cone
The ratio of the base radius to the height of the cone is given as \( 8:15 \).
Let's denote the base radius by \( r \) and the height by \( h \). Since the ratio is \( 8:15 \):
[tex]\[ \frac{r}{h} = \frac{8}{15} \][/tex]
[tex]\[ h = \frac{15}{8} r \][/tex]
Substituting \( r \approx 6.96 \):
[tex]\[ h \approx \frac{15}{8} \times 6.96 \][/tex]
[tex]\[ h \approx 13.04 \, \text{m} \][/tex]
### Step 3: Determine the Slant Height of the Cone
To find the slant height \( l \) of the cone, we can use the Pythagorean theorem in the context of the cone's dimensions. The slant height \( l \) is given by:
[tex]\[ l = \sqrt{r^2 + h^2} \][/tex]
Substituting \( r \approx 6.96 \) and \( h \approx 13.04 \):
[tex]\[ l \approx \sqrt{6.96^2 + 13.04^2} \][/tex]
[tex]\[ l \approx \sqrt{48.42 + 170.04} \][/tex]
[tex]\[ l \approx \sqrt{218.46} \][/tex]
[tex]\[ l \approx 14.78 \, \text{m} \][/tex]
### Step 4: Determine the Curved Surface Area of the Cone
The formula for the curved surface area of a cone is:
[tex]\[ \text{Curved Surface Area} = \pi r l \][/tex]
Substituting \( r \approx 6.96 \) and \( l \approx 14.78 \):
[tex]\[ \text{Curved Surface Area} \approx \pi \times 6.96 \times 14.78 \][/tex]
[tex]\[ \text{Curved Surface Area} \approx 323.00 \, \text{m}^2 \][/tex]
### Step 5: Determine the Volume of the Cone
The formula for the volume of a cone is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Substituting \( r \approx 6.96 \) and \( h \approx 13.04 \):
[tex]\[ V \approx \frac{1}{3} \pi \times (6.96)^2 \times 13.04 \][/tex]
[tex]\[ V \approx \frac{1}{3} \pi \times 48.42 \times 13.04 \][/tex]
[tex]\[ V \approx \frac{1}{3} \times \pi \times 631.89 \][/tex]
[tex]\[ V \approx 660.80 \, \text{m}^3 \][/tex]
### Summary of Results:
- The base radius \( r \) of the cone: \( 6.96 \, \text{m} \)
- The height \( h \) of the cone: \( 13.04 \, \text{m} \)
- The slant height \( l \) of the cone: \( 14.78 \, \text{m} \)
- The curved surface area of the cone: \( 323.00 \, \text{m}^2 \)
- The volume of the cone: [tex]\( 660.80 \, \text{m}^3 \)[/tex]
