A man is in a boat 2 miles from the nearest point on the coast. He is to go to p

Question

A man is in a boat 2 miles from the nearest point on the coast. He is to go to point Q, located 3 miles down the coast and 1 mile inland. He can row at a rate of 3 miles per hour and walk at 3 miles per hour. Toward what point on the coast should he row in order to reach point Q in the least time? (Round your answer to two decimal places.)

_________ mile(s) down the coast

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 6 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area. (Round your answer to three decimal places.)

_________ cm

Explanation / Answer

The surface area of given solid

S = 2*pi*r*h + 2*pi*r^2 + 2*pi*r^2
S = 2*pi*r*h + 4*pi*r^2

The volume of the solid:

V = pi*r^2*h + (2/3)*pi*r^3 + (2/3)*pi*r^3
V = pi*r^2*h + (4/3)*pi*r^3
6 = pi*r^2*h + (4/3)*pi*r^3
h = [6 - (4/3)*pi*r^3] / [pi*r^2]

Sub that into the surface area formula:

SA = 2*pi*r*[6 - (4/3)*pi*r^3] / [pi*r^2] + 4*pi*r^2
SA = (2/r)*(6-(4/3*pi*r^3)) + 4*pi*r^2

= (12/r) - (8/3)*pi*r^2 + 4*pi*r^2

= (12/r) + ((-8/3)+(4))*pi*r^2

= (12/r) + (4/3)*pi*r^2

for minimum surface area

d(SA)/dr = 0

d/dr[(12/r)+4/3*pi*r^2] = 0

(-12/r^2) + (4/3*pi*2*r) = 0

(8*pi*r/3) = 12/r^2

(8*pi*r^3) = 36

r^3 = (36/8*pi)

r = 1.127 cm

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.