Find the area of the following shapes after the transformations have been made.
A circle has an area of 11. If the radius is increased by a factor of 3, what is the new area of the circle?
Original area
original Area = 11 = π (original radius)2
New area
new Area = π (new radius)2
From the statement the radius is increased by a factor of 3, we get
new radius = 3 × original radius
| new Area |
= |
π (new radius)2 |
| |
= |
π (3 × original radius)2 |
| |
= |
9 × π (original radius)2 |
| |
= |
9 × original Area |
| |
= |
9 × 11 |
| |
= |
99 |
Alternate solution:
The radius increased by 3. In the circle area equation, the radius is squared so we can square the factor 3 and multiply that by the original area to get 32 × 11 = 99.
The answer is
Find the volume of the following shapes after the transformations have been made.
A triangular prism has a volume of 20. If the base height is increased by a factor of 4, what is the new volume of the triangular prism?
Original volume
original Volume = 20 = (1/2)(base)(original height)(length)
New volume
new Volume = (1/2)(base)(new height)(length)
From the statement the base height is increased by a factor of 4, we get
new height = 4 × old height
| new Volume |
= |
(1/2)(base)(new height)(length) |
| |
= |
(1/2)(base)(4 × old height)(length) |
| |
= |
4 × (1/2)(base)(old height)(length) |
| |
= |
4 × original Volume |
| |
= |
4 × 20 |
| |
= |
80 |
Alternate solution:
The base height increased by 4. In the triangular prism volume formula, the base height is linear so we can multiply the increased factor by the original volume to get 4 × 20 = 80.
The answer is
