MHB Solve Scale Factor Problem: 'ET Pizza' Pizzas

[tantrik]

Dear friends,

I am unable to solve the following scale factor problem. Will appreciate your help here. Thanks in advance.

'ET Pizza' produces two pizzas that are similar in shape. The smaller pizza is 20 cm in diameter and costs $10. The larger pizza is 30 cm in diameter. What is a fair cost for the larger pizza?

The answer in the book is $33.75 but I am getting $15 (scale factor = 30/20=1.5; cost of larger pizza = $10*1.5 = $15). Let me know where I am mistaken.

 

[MarkFL]

A pizza is a 3-dimensional object. If the 2 pizzas are similar in shape, then increasing a linear measure in the smaller by some factor $k$ will result in the volume of the larger increasing by $k^3$. And so we would find the fair price $P$ of the larger pizza to be:

$$P=\left(\frac{30}{20}\right)^310=\frac{135}{4}=33.75$$

You see the larger pizza, being 1.5 times larger in all 3 spatial dimensions (making it similar in shape to the smaller), has 3.375 times as much volume.

 

[tantrik]

A pizza is a 3-dimensional object. If the 2 pizzas are similar in shape, then increasing a linear measure in the smaller by some factor $k$ will result in the volume of the larger increasing by $k^3$. And so we would find the fair price $P$ of the larger pizza to be:

$$P=\left(\frac{30}{20}\right)^310=\frac{135}{4}=33.75$$

You see the larger pizza, being 1.5 times larger in all 3 spatial dimensions (making it similar in shape to the smaller), has 3.375 times as much volume.

Thanks for the solution