Final answer:
To find the volume of the pyramid, we need to find the radius of the larger cone. We can set up an equation using the surface areas of the two cones and solve for the radius. Once we have the radius, we can substitute it into the volume formula to find the volume of the pyramid.
Step-by-step explanation:
To find the volume of the pyramid, we first need to find the radius of the larger cone. Since the surface area of the larger cone is 65π square inches, we can set up the equation:
65π = πr²
Simplifying the equation, we get
r² = 65
Next, we need to find the ratio of the surface areas of the two cones:
65π/41.6π = r²/6.4²
Simplifying the equation, we get:
r² = 6.4² × (65/41.6)
Now, we can substitute the value of r² into the volume formula for a cone:
V = (1/3)πr²h
We can calculate the volume using the given values of the radius of the smaller cone and the height of the larger cone:
V = (1/3)π(6.4²) × h
Finally, substitute the known value of the larger cone's surface area to solve for h:
65π = π(6.4²) + π × r × sqrt(r² + h²)
Solving for h, we can substitute the value of h into the volume formula to find the volume of the pyramid.