Warrior Cups

In: Computers and Technology

Submitted By ryangillis
Words 355
Pages 2
Warrior Cups Inc.

Cylinder: V= πr²h
SA= 2πrh + 2πr²
Half-Sphere : V= 2/3πr³
SA= 2πr²

V= 900 cm³
Lid Plastic: ¢0.10/cm²
Cup Plastic: ¢0.05/cm²

r represents the radius h represents the height

Find h in terms of r:
900 = πr² + 2/3 πr³
900 = πr²(h + 2/3r)
900/ πr² = h + 2/3r
(900/ πr²) – 2/3r = h Cost in terms of r :
C(r) = 0.05(2 πrh + πr²) + 0.10(2 πr²) = 0.1 πrh + 0.05 πr² +0.2 πr² = 0.1 πr(900/ πr² - 2r/3) + 0.25 πr² *insert h = 90 πr/ πr² - 0.2 πr²/3 + 0.25 πr² = 90/r + 1.1 πr²/6
C’(r) = -90/r² + 1.1 πr/3 * take derivative 0 = -90/r² + 1.1 πr/3 * set derivative equal to zero r²(0) = (-90/r² + 1.1 πr/3) r² *multiply both sides by r² 0 = -90r²/r² + 1.1 πr³/3 = -90 + 1.1 πr³/3 r = (90/1.1 π)^1/3 r ≈ 4.27cm Therefore, the radius is approximately 4.27cm

Double derivative Check:
C’’(r) = 1.1 πr²
C’’(4.27) = +ve
Therefore it is a minimum

Finding height: h = 900/ π(4.27)² - 2/3(4.27)*plug the radius into the height equation h = 12.87
Therefore the height is 12.87cm

Dimensions and Cost: * r = 4.27cm * h = 12.87 cm * V = 900 cm³ * Lid Plastic: ¢0.10/cm² * Cup Plastic: ¢0.05/cm² Demand function:
X = 100,000 -5,000n n = 100,000 – x/5,000 *Isolate for n variable
P = 0.99 + 0.10n
P(x) = 0.99 + 0.10(100,000 – x/5,000) *plug in n value = 0.99 + 2 – 0.00002x = 2.99 – 0.00002x
X = # of $0.10 increases

Revenue Function:
R(x) = xp(x)
R(x) = 2.99x – 0.00002x²
R’(x) = 2.99 – 0.00004x

Cost Function:
C = -0.0000004x² + 0.316x + 2000
C’ = -0.0000008x² + 0.316

Profit Function:
P = (2.99x – 0.00002x²) – (-0.0000004x² + 0.316x + 2000) = -0.0000196x² + 2.674x – 2000
P’ = -0.0000392x +…...

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