Two cones are displayed. The smaller cone to the left is labeled 'r equals 8 feet' and the slant height is labeled '12 feet.' The larger cone to the right is labeled below with an x and the slant height is labeled '15 feet.'
What is the length of the radius of the larger cone?
A. 10 ft
B. 11 ft
C. 12 ft
D. 13 ft
assuming the cones are similar (otherwise there is no way to determine x),
x/8 = 15/12
x = 10
so, A
To find the length of the radius of the larger cone, we can use the concept of similar triangles.
Since the smaller cone on the left has a radius of 8 feet and a slant height of 12 feet, we can set up the following proportion:
(radius of larger cone) / (slant height of larger cone) = (radius of smaller cone) / (slant height of smaller cone)
Let's substitute the given values into the equation:
(x) / (15 ft) = (8 ft) / (12 ft)
Now, let's cross-multiply and solve for x:
12x = 8 * 15
12x = 120
x = 120 / 12
x = 10 ft
Therefore, the length of the radius of the larger cone is 10 ft, which means the correct answer is A. 10 ft.
We can use the formula for the slant height of a cone: $l=\sqrt{r^2+h^2}$, where $r$ is the radius and $h$ is the height. We can set up an equation using this formula for each cone:
For the smaller cone: $12=\sqrt{8^2+h^2}$
Solving for $h$, we get $h=4\sqrt{5}$
For the larger cone: $15=\sqrt{x^2+h^2}$
Substituting the value we found for $h$, we get $15=\sqrt{x^2+80}$
Squaring both sides gives $225=x^2+80$
Solving for $x$, we get $x=\pm 13$
Since $x$ represents a length, we take the positive value, so the radius of the larger cone is $\boxed{\textbf{(D)}\ 13 \ \text{ft}}$.