To solve this problem, we need to find the slant height of a cone given the following conditions:
1. The ratio of the radius to the slant height of the cone is [tex]\(3:5\)[/tex].
2. The total surface area of the cone is [tex]\(\frac{2112}{7}\)[/tex] square cm.
Let's denote:
- The radius of the cone as [tex]\( r \)[/tex].
- The slant height of the cone as [tex]\( l \)[/tex].
Given the ratio, we can express [tex]\( r \)[/tex] and [tex]\( l \)[/tex] as:
[tex]\[ r = 3k \][/tex]
[tex]\[ l = 5k \][/tex]
where [tex]\( k \)[/tex] is a constant.
The formula for the total surface area of a cone is:
[tex]\[ \text{Total Surface Area} = \pi r (r + l) \][/tex]
Substituting [tex]\( r \)[/tex] and [tex]\( l \)[/tex] with [tex]\( 3k \)[/tex] and [tex]\( 5k \)[/tex] respectively, we get:
[tex]\[ \text{Total Surface Area} = \pi (3k) \left(3k + 5k\right) \][/tex]
[tex]\[ \text{Total Surface Area} = \pi (3k) (8k) \][/tex]
[tex]\[ \text{Total Surface Area} = \pi (24k^2) \][/tex]
We know the total surface area is given as [tex]\(\frac{2112}{7}\)[/tex] square cm:
[tex]\[ \pi (24k^2) = \frac{2112}{7} \][/tex]
To find [tex]\( k \)[/tex], we simplify and solve the equation:
[tex]\[ \pi (24k^2) = 301.7142857142857 \][/tex]
Solving for [tex]\( k^2 \)[/tex]:
[tex]\[ 24\pi k^2 = 301.7142857142857 \][/tex]
[tex]\[ k^2 = \frac{301.7142857142857}{24\pi} \][/tex]
[tex]\[ k = \pm \sqrt{\frac{301.7142857142857}{24\pi}} \][/tex]
Given that [tex]\( k \)[/tex] can take a positive or negative value, and we are dealing with physical dimensions, we use the positive value of [tex]\( k \)[/tex]:
[tex]\[ k \approx 2.00040245894147 \][/tex]
Given this value of [tex]\( k \)[/tex], we can now find the slant height [tex]\( l \)[/tex]:
[tex]\[ l = 5k \][/tex]
[tex]\[ l = 5 \times 2.00040245894147 \][/tex]
[tex]\[ l \approx 10.00201229470736 \][/tex]
Therefore, the slant height of the cone is approximately [tex]\( 10.002 \)[/tex] cm.
