Answer: Your friends pizza is larger
Step-by-step explanation:
Choice of Pizza
by: Staff
Answer
Part I
Obviously your friend’s pizza is larger than your pizza.
We assume the pizzas are formed in the traditional round shape, so the radius of your friend’s pizza is larger than the radius of your pizza.
The area of a pizza is:
APizza = πr²
The area of your pizza
Ayou = area of pizza eaten by you
ryou = radius of pizza eaten by you
Ayou = πr²you
The area of your friend’s pizza
Afriend = area of pizza eaten by friend
rfriend = radius of pizza eaten by friend
Afriend = πr²friend
At what radius will the one-third the area of the friend’s pizza exceed one-half the area of your pizza?
⅓ * πr²friend > ½ * πr²you
Solve for rfriend
Multiply each side of the inequality by 3
3 * (⅓) * πr²friend > 3 * (½) * πr²you
1 * πr²friend > 3/2 * πr²you
πr²friend > 3/2 * πr²you
Divide each side of the inequality by π
πr²friend / π > 3/2 * πr²you / π
r²friend * (π/ π) > 3/2 * r²you * (π/ π)
r²friend * 1 > 3/2 * r²you * 1
r²friend > 3/2 * r²you
Take the square root of each side of the inequality
√(r²friend) > √(3/2 * r²you)
rfriend > √(3/2) * √(r²you)
rfriend > 1.2247448713916 * ryou
rfriend > 1.225 * ryou
If the radius of your friend’s pizza is over 22.5% larger than your pizza, then ⅓ of your friend’s pizza will be greater than ½ of your pizza.
For example, if your pizza has a radius of 5 inches, and your friend’s pizza has a radius of 6.2 inches, then ⅓ of your friend’s pizza will be greater than ½ of your pizza.
⅓ * πr²friend > ½ * πr²you
⅓ * π * (6.2)² > ½ * π * (5)²
⅓ * π * (38.44) > ½ * π * (25)
40.2542738679972 in² > 39.2699081698724 in²
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Dec 12, 2012
Choice of Pizza
by: Staff
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Part II
Math – Example - ⅓ of a larger pizza is greater than ½ of a smaller pizza.
