Question

find the difference between the two polynomials to solve a real-world problem. the models needed for each situation are provided.

1. darnell and stephanie have competing refreshment stand businesses. stephanies profit can be modeled with the polynomial $2c^{2}-7c-200$, where $c$ is the number of items sold. darnells profit can be modeled with the polynomial $c^{2}+8c-100$.

(a) write a polynomial that represents the difference between stephanies profit and darnells profit.
(b) find this difference when the number of items sold, $c$, is 30.

1. the number of gallons of water in a leaking pool is determined by the rate that the water is filling, $8g^{2}+3g-4$, and the rate that water leaks from the pool, $9g^{2}-2g-5$, where $g$, represents the number of gallons entering or leaving the pool per minute.

(a) write an expression for the net change in gallons per minute of the water in the pool.
(b) find the change in the amount when the rate, $g$, is 5 gallons per minute.

Explanation:

Problem 1 (a)

Step1: Set up the difference polynomial

$\text{Stephanie's profit} - \text{Darnell's profit} = (2c^2 - 7c - 200) - (c^2 + 8c - 100)$

Step2: Distribute the negative sign

$2c^2 - 7c - 200 - c^2 - 8c + 100$

Step3: Combine like terms

$(2c^2 - c^2) + (-7c - 8c) + (-200 + 100) = c^2 - 15c - 100$

Problem 1 (b)

Step1: Substitute $c=30$ into the polynomial

$\text{Difference} = (30)^2 - 15(30) - 100$

Step2: Calculate each term

$900 - 450 - 100$

Step3: Compute the final value

$900 - 450 - 100 = 350$

Problem 2 (a)

Step1: Set up net change expression

$\text{Filling rate} - \text{Leaking rate} = (8g^2 + 3g - 4) - (9g^2 - 2g - 5)$

Step2: Distribute the negative sign

$8g^2 + 3g - 4 - 9g^2 + 2g + 5$

Step3: Combine like terms

$(8g^2 - 9g^2) + (3g + 2g) + (-4 + 5) = -g^2 + 5g + 1$

Problem 2 (b)

Step1: Substitute $g=5$ into the expression

$\text{Net change} = -(5)^2 + 5(5) + 1$

Step2: Calculate each term

$-25 + 25 + 1$

Step3: Compute the final value

$-25 + 25 + 1 = 1$

Answer:

1. (a) $c^2 - 15c - 100$
2. (b) $350$
3. (a) $-g^2 + 5g + 1$
4. (b) $1$