Question

the student photo club at the college is planning on selling prints that it makes to raise money.
the profit p, in dollars, from selling x prints is given by the function:
$p(x) = 189x - 2x^2$

a) find the number of prints, to the nearest whole print, that need to be sold to maximize the profit.
you must sell \boxed{} prints to maximize the profit.

b) the maximum profit, to the nearest dollar, is $\boxed{}$. (no dollar signs or commas.)

Explanation:

Step1: Identify the function type

The profit function \( P(x) = 189x - 2x^2 \) is a quadratic function in the form \( ax^2 + bx + c \) (here \( a=-2 \), \( b = 189 \), \( c = 0 \)). For a quadratic function, if \( a<0 \), the parabola opens downward, and the vertex is the maximum point. The x - coordinate of the vertex of a quadratic function \( y=ax^2+bx + c \) is given by \( x=-\frac{b}{2a} \).

Step2: Calculate the number of prints for maximum profit

Substitute \( a=-2 \) and \( b = 189 \) into the formula \( x=-\frac{b}{2a} \).
\( x=-\frac{189}{2\times(-2)}=\frac{189}{4} = 47.25 \). Rounding to the nearest whole number, we get \( x = 47 \) (or 47 when rounded, since 0.25 is closer to 0 in terms of whole number? Wait, no, 47.25, the nearest whole number is 47? Wait, 47.25 is 47 and a quarter, so the nearest whole number is 47? Wait, actually, when dealing with the number of prints, we can also consider that the vertex is at \( x = 47.25 \), so the nearest whole number is 47 or 48? Wait, let's check. The formula for the vertex of \( ax^2+bx + c \) is \( x=-\frac{b}{2a} \). So \( a=-2 \), \( b = 189 \), so \( x=\frac{189}{4}=47.25 \). So to the nearest whole print, it's 47 (since 0.25 is less than 0.5, we round down? Wait, but in some cases, for maximum, even if it's 47.25, the actual maximum occurs at 47.25, so the nearest whole number is 47 or 48? Wait, let's calculate \( P(47) \) and \( P(48) \).

\( P(47)=189\times47-2\times47^2=189\times47 - 2\times2209=8883-4418 = 4465 \)

\( P(48)=189\times48-2\times48^2=9072-2\times2304=9072 - 4608=4464 \)

\( P(47.25)=189\times47.25-2\times(47.25)^2=189\times47.25-2\times2232.5625=8930.25 - 4465.125 = 4465.125 \)

So the maximum is at 47.25, and the nearest whole number is 47 (since \( P(47)=4465 \) and \( P(48)=4464 \), so 47 gives a higher profit than 48? Wait, no, wait 4465 is higher than 4464, so 47 is better. Wait, but 47.25 is the exact vertex. So the number of prints to maximize profit, to the nearest whole print, is 47 (or 47.25 rounded to nearest whole is 47).

Step3: Calculate the maximum profit

Now, to find the maximum profit, we substitute \( x = 47.25 \) into the profit function \( P(x) \).

\( P(47.25)=189\times47.25-2\times(47.25)^2 \)

First, calculate \( 189\times47.25 \):

\( 189\times47=189\times(40 + 7)=189\times40+189\times7 = 7560+1323 = 8883 \)

\( 189\times0.25 = 47.25 \)

So \( 189\times47.25=8883 + 47.25=8930.25 \)

Then, \( 2\times(47.25)^2=2\times(2232.5625)=4465.125 \)

So \( P(47.25)=8930.25-4465.125 = 4465.125 \). Rounding to the nearest dollar, it's 4465.

Wait, but let's check with \( x = 47 \):

\( P(47)=189\times47-2\times47^2=8883-2\times2209=8883 - 4418=4465 \)

With \( x = 48 \):

\( P(48)=189\times48-2\times48^2=9072-2\times2304=9072 - 4608=4464 \)

So the maximum profit at \( x = 47 \) is 4465, and at \( x = 48 \) is 4464, so the maximum profit is 4465, and the number of prints is 47 (since 47.25 is closer to 47 in terms of the value that gives the maximum profit, as seen from the calculations of \( P(47) \) and \( P(48) \)).

Answer:

(for part a):
47