Two cones are similar. The small one has a radius of 3cm and the big one has a radius of 4cm. What is the scale factor of:
1. Their heights
2. Their base areas
3. Their volumes
1. The scale factor of their heights can be found by dividing the height of the larger cone by the height of the smaller cone:
Height of larger cone = h1
Height of smaller cone = h2
Using the formula for similar triangles:
h1 / h2 = r1 / r2
h1 / h2 = 4 / 3
h1 / h2 = 4:3
Therefore, the scale factor of their heights is 4:3.
2. The scale factor of their base areas can be found by dividing the area of the base of the larger cone by the area of the base of the smaller cone:
Area of larger base = A1
Area of smaller base = A2
Using the formula for the area of a circle:
A1 = π * r1^2
A2 = π * r2^2
A1 / A2 = (π * r1^2) / (π * r2^2)
A1 / A2 = (4^2) / (3^2)
A1 / A2 = 16 / 9
Therefore, the scale factor of their base areas is 16:9.
3. The scale factor of their volumes can be found by dividing the volume of the larger cone by the volume of the smaller cone:
Volume of larger cone = V1
Volume of smaller cone = V2
Using the formula for the volume of a cone:
V1 = (1/3) * π * r1^2 * h1
V2 = (1/3) * π * r2^2 * h2
V1 / V2 = [(1/3) * π * r1^2 * h1] / [(1/3) * π * r2^2 * h2]
V1 / V2 = (r1^2 * h1) / (r2^2 * h2)
V1 / V2 = (4^2 * h1) / (3^2 * h2)
V1 / V2 = (16 * h1) / (9 * h2)
V1 / V2 = 16 / 9 * h1 / h2
Since we already found that the scale factor of their heights is 4:3, we can substitute this ratio:
V1 / V2 = 16 / 9 * 4 / 3
V1 / V2 = 64 / 27
Therefore, the scale factor of their volumes is 64:27.