Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
### 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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
