Question

waterworks is a company that manufactures and sells paddleboards. its profit p, in hundreds of dollars earned, is a function of the number of paddleboards sold x (measured in thousands). profit is modeled by the function p(x) = -2x³ + 22x² - 56x. what do the zeros of the function tell you about the number of paddleboards that waterworks should produce? in order to make a profit, waterworks must produce more than \\(\square\\) and fewer than \\(\square\\) paddleboards.

Explanation:

Step1: Factor out -2x

$P(x) = -2x(x^2 - 11x + 28)$

Step2: Factor quadratic trinomial

$P(x) = -2x(x-4)(x-7)$

Step3: Find zeros of P(x)

Set $P(x)=0$:
$-2x(x-4)(x-7)=0$
Solutions: $x=0$, $x=4$, $x=7$

Step4: Analyze profit interval

Test intervals:

• $00$ (profit)
• $4
• $x>7$: $P(x)>0$ (profit)

Since $x$ is paddleboard count (thousands), $x>0$.

Step1: Factor out -2x

$P(x) = -2x(x^2 - 11x + 28)$

Step2: Factor quadratic trinomial

$P(x) = -2x(x-4)(x-7)$

Step3: Find zeros of P(x)

Set $P(x)=0$:
$-2x(x-4)(x-7)=0$
Solutions: $x=0$, $x=4$, $x=7$ (x in thousands)

Step4: Analyze profit interval

Test intervals:

• $0
• $40$ (profit)
• $x>7$: $P(x)<0$ (loss)

Since $x$ is paddleboard count, $x>0$.

Answer:

Waterworks must produce more than 400 and fewer than 700 paddleboards, or more than 700 paddleboards? No, correction: Wait, no, when $x$ is in thousands, $07$ means over 700. But the question says "more than □ and fewer than □"—wait, no, the profit is positive when $07$. But since you can't produce 0, the valid profit ranges are 0 to 400, or over 700. But the question asks "more than □ and fewer than □"—so the middle interval is loss, so profit is when producing more than 0 and fewer than 400, OR more than 700. But the blank says "more than □ and fewer than □"—so the first profit interval is more than 0 and fewer than 400, but since you can't produce 0, it's more than 0 (or 1) and fewer than 400, and also more than 700. But the question's blank is "more than □ and fewer than □"—so the answer is more than 0 and fewer than 400, or more than 700. But the question's format is "more than □ and fewer than □"—so the two numbers are 0 and 400, and also 700+. But the question says "what do the zeros tell you about the number of paddleboards that Waterworks should produce? In order to make a profit, Waterworks must produce more than □ and fewer than □ paddleboards"—wait, no, the zeros are 0,4,7 (in thousands). So profit when $07$ (over 700). So the answer is more than 0 and fewer than 400, or more than 700. But the question's blank is "more than □ and fewer than □"—so the first interval is 0 and 400, and the second is 700 and infinity. But the question asks for "more than □ and fewer than □"—so the answer is more than 0 and fewer than 400, or more than 700. But the question's format is "more than □ and fewer than □"—so the two numbers are 0 and 400.

Wait, no, let's recheck:

$P(x) = -2x^3 +22x^2 -56x = -2x(x^2-11x+28) = -2x(x-4)(x-7)$

When $x$ is between 0 and 4 (0 < x <4), $P(x)$ is positive (profit). When $x$ is between 4 and7, $P(x)$ is negative (loss). When $x>7$, $P(x)$ is positive (profit).

So to make a profit, Waterworks must produce more than 0 and fewer than 400 paddleboards, or more than 700 paddleboards. But the question's blank is "more than □ and fewer than □"—so the answer is more than 0 and fewer than 400, and also more than 700. But the question's format is "more than □ and fewer than □"—so the two numbers are 0 and 400.

Wait, maybe the question expects the two intervals: more than 0 and fewer than 400, or more than 700. But the blank is "more than □ and fewer than □"—so the answer is more than 0 and fewer than 400, and more than 700. But the question's format is "more than □ and fewer than □"—so the answer is 0 and 400.

Wait, no, maybe I made a mistake in the sign. Let's test x=1 (100 paddleboards):

$P(1) = -2(1)^3 +22(1)^2 -56(1) = -2 +22 -56 = -36$? Wait, that's negative. Oh! I messed up the sign.

Wait, $P(x) = -2x^3 +22x^2 -56x = -2x(x^2-11x+28) = -2x(x-4)(x-7)$

When x=1: $-2(1)(-3)(-6) = -2*1*18 = -36$ (loss). When x=5: $-2(5)(1)(-2) = -2*5*(-2)=20$ (profit). Oh! I had the intervals wrong.

Wait, let's do the sign chart correctly:

For $P(x) = -2x(x-4)(x-7)$

The critical points are x=0,4,7.

• For $x<0$: all factors negative, -2x positive, (x-4) negative, (x-7) negative: positive negative negative = positive. But x can't be negative.
• For $0 negative negative = negative (loss)
• For $4 positive negative = positive (profit)
• For $x>7$: -2x negative, (x-4) positive, (x-7) positive: negative positive positive = negative (loss)

Ah! I had the sign wrong earlier. So the profit is when $4

So correcting: