You have been put in charge of making pizza for the fundraiser. You are told to make the larger pizzas with a radius 3 less than twice the radius of the smaller pizzas. The difference in the area of the pizzas is to 24 pi. Find the radius of the smaller pizza.
A = π r^2
Difference (24π) = (2r-3)^2π - r^2π
Divide both sides by π .
24 = (2r-3)^2 - r^2
I'll let you calculate the rest.
r^2=1
To solve this problem, let's break it down step by step.
Step 1: Assign variables
Let's call the radius of the smaller pizza "r". We are told that the larger pizzas have a radius that is 3 less than twice the radius of the smaller pizza. So, the radius of the larger pizza can be represented as "2r - 3".
Step 2: Find the area of the pizzas
The area of a circle can be calculated using the formula A = πr^2. For the smaller pizza, the area would be A1 = πr^2. For the larger pizza, the area would be A2 = π(2r - 3)^2.
Step 3: Apply the given information
We are told that the difference in the area of the pizzas is 24π. This can be expressed as A2 - A1 = 24π.
Step 4: Substitute the area equations into the equation
Replace A1 with πr^2 and A2 with π(2r - 3)^2 in the equation A2 - A1 = 24π:
π(2r - 3)^2 - πr^2 = 24π
Simplifying the equation:
(2r - 3)^2 - r^2 = 24
Expand and simplify:
4r^2 - 12r + 9 - r^2 = 24
3r^2 - 12r - 15 = 0
Step 5: Solve the quadratic equation
To find the radius of the smaller pizza, we need to solve the quadratic equation 3r^2 - 12r - 15 = 0.
Factoring the equation:
(3r + 3)(r - 5) = 0
Setting each factor equal to zero and solving for r:
3r + 3 = 0 -> 3r = -3 -> r = -1
r - 5 = 0 -> r = 5
Since the radius cannot be negative in this context, the radius of the smaller pizza is 5 units.