Question

carrie is considering three different types of materials for her chairs. depending on the price she sets for each chair, the costs and revenues for each type of material will generate different costs and revenues for her business. in the functions in the table, x is the price of each chair. the functions give the revenue (r) and costs (c) that carries business can expect in a year at a given price for the item. use graphing tools to examine the graphs for each set of equations. then select the correct answer from each drop - down menu. table with material 1: (r(x)=200000x - 2000x^{2}), (c(x)=5000000 - 20000x); material 2: (r(x)=160000x - 1000x^{2}), (c(x)=4000000 - 10000x); material 3: (r(x)=54000x - 250x^{2}), (c(x)=2000000 - 5000x) material 1 can deliver the highest profit of about $2,053,000 (drop - down with options $500,000, $2,055,000, $3,225,000, $4,125,000) material blank earns profit for the widest range of pri

Explanation:

Step1: Define profit function

Profit $P(x) = R(x) - C(x)$

Step2: Calculate Material 1 profit

$P_1(x) = (200000x - 2000x^2) - (5000000 - 20000x) = -2000x^2 + 220000x - 5000000$
Find vertex: $x = -\frac{b}{2a} = -\frac{220000}{2(-2000)} = 55$
$P_1(55) = -2000(55)^2 + 220000(55) - 5000000 = 1,050,000$

Step3: Calculate Material 2 profit

$P_2(x) = (160000x - 1000x^2) - (4000000 - 10000x) = -1000x^2 + 170000x - 4000000$
Find vertex: $x = -\frac{170000}{2(-1000)} = 85$
$P_2(85) = -1000(85)^2 + 170000(85) - 4000000 = 3,225,000$

Step4: Calculate Material 3 profit

$P_3(x) = (54000x - 270x^2) - (2000000 - 5000x) = -270x^2 + 59000x - 2000000$
Find vertex: $x = -\frac{59000}{2(-270)} \approx 109.26$
$P_3(109.26) \approx -270(109.26)^2 + 59000(109.26) - 2000000 \approx 1,271,605$

Step5: Find profit ranges (solve $P(x)>0$)

Material 1: Solve $-2000x^2 + 220000x - 5000000 > 0$
Roots: $x = \frac{-220000 \pm \sqrt{220000^2 - 4(-2000)(-5000000)}}{2(-2000)} \approx 25, 100$
Range: $25 < x < 100$ (width=75)
Material 2: Solve $-1000x^2 + 170000x - 4000000 > 0$
Roots: $x = \frac{-170000 \pm \sqrt{170000^2 - 4(-1000)(-4000000)}}{2(-1000)} \approx 25, 145$
Range: $25 < x < 145$ (width=120)
Material 3: Solve $-270x^2 + 59000x - 2000000 > 0$
Roots: $x = \frac{-59000 \pm \sqrt{59000^2 - 4(-270)(-2000000)}}{2(-270)} \approx 40.1, 178.4$
Range: $40.1 < x < 178.4$ (width≈138.3)

Answer:

1. Material 2 can deliver the highest profit of about $\$3,225,000$
2. Material 3 earns profit for the widest range of prices