Question

the radius of a large pizza is 1 in less than twice the radius of a small pizza. the difference between the areas of the two pizzas is 33pi in^2. find the radius of the large pizza.

Answer 1

It might make it easier to draw a picture. Saying r1=radius of small pizza and r2=radius of large pizza, A1=area of small, A2=area of large. Remember the formula for the area of a circle:\[Area = \pi r^2\]And we know that \[2 \times r1 = r2 - 1\]because r2 is 1" less than two times r2.
We also know, \[A2-A1=33\pi\]
Now using the area formula with A1 and A2, we get \[\pi r2^2 - \pi r1^2 = 33 \pi\]
We can factor pi out of both sides and solve for r2, we get\[ r1=\sqrt{-33+r2^2}\]
Now put that into the other equation to get \[2\sqrt{-33+r2^2}=r2-1\]
Square both sides and solve for r2. And you have your answer.

Answer 2

Let R and r be the radius of the large and small pizza respectively. Then from the problem statement:\[\pi R^2-\pi r^2 =33\pi\]and\[R=2r-1 \text{ or } r=\frac{1+R}{2} \]Replacing r in the first equation above with (1+R)/2 yields\[\pi R^2-\frac{1}{4} \pi (1+R)^2=33 \pi\]Solve the above for R and select the positive solution.

R = 7 inches