### Instruction: P.J. has a cylindrical mug with a 3-inch diameter. His mug is filled to a height of 6 inches with grape juice. Schuyler has a cylindrical mug with a 4-inch diameter. To what height in inches must Schuyler's mug be filled so that he receives the same amount of juice as P.J.? Express your answer as a mixed number.### Response: I need to compare the volumes of the two cylindrical mugs, since volume is a measure of how much juice they can hold.
I recall that the formula for the volume of a cylinder is V = pi*r^2*h, where r is the radius and h is the height.
For P.J.
's mug, the radius is half of the diameter, so r = 3/2 inches.
The height is given as 6 inches.
So the volume of P.J.
's mug is V = pi*(3/2)^2*6.
I can simplify this expression by multiplying the numbers inside the parentheses.
(3/2)^2 = 9/4, and 9/4*6 = 27/2.
So the volume of P.J.
's mug is V = 27/2*pi cubic inches.
For Schuyler's mug, the radius is also half of the diameter, so r = 4/2 = 2 inches.
The height is unknown, so I'll call it x inches.
So the volume of Schuyler's mug is V = pi*2^2*x.
I can simplify this expression by squaring 2.
2^2 = 4, so the volume of Schuyler's mug is V = 4*pi*x cubic inches.
To find the height x, I need to set the volumes of the two mugs equal, since the problem says they receive the same amount of juice.
So I have the equation 27/2*pi = 4*pi*x.
I can solve for x by dividing both sides by 4*pi.
This cancels out the pi terms and gives me x = (27/2*pi)/(4*pi) = 27/8.
So Schuyler's mug must be filled to a height of 27/8 inches.
The problem asks me to express the answer as a mixed number, which means a whole number plus a fraction.
To do this, I need to divide 27 by 8 and write the quotient and remainder as a mixed number.
I know that 8 goes into 27 three times, with a remainder of 3.
So 27/8 = 3 + 3/8.
Therefore, the final answer is 3 and 3/8 inches.