1) A box has a bottom with one edge 6 times as long as the other. If the box has

Question

1)  A box has a bottom with one edge 6 times as long as the other. If the box has no top and the volume is fixed at V, what dimensions minimize the surface area?

dimensions =  
Enter the dimensions as a comma-separated list, e.g., 3,sqrt(12),8. (Your answer may involve V.)

2) A small island is 3 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the boat be landed in order to arrive at a town 8 miles down the shore from P in the least time? Let x be the distance (in miles) between point P and where the boat lands on the lakeshore.

(a) Enter a function T(x) that describes the total amount of time the trip takes as a function of the distance x.
T(x)=  (include units)

(b) What is the distance x=c that minimizes the travel time?
c=  (include units)

(c) What is the least travel time?
The least travel time is (include units)

3) A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=11x2. What are the dimensions of such a rectangle with the greatest possible area?

Width =  
Height =

4) An oil refinery is located on the north bank of a straight river that is 2km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 8km east of the refinery. The cost of laying pipe is $300,000/km over land to point P on the north bank and $600,000/km under river to the tanks. What is the minimal cost to construct a pipeline from the refinery to the storage tanks?

Explanation / Answer

1) let one edge of the base =x

one edge 6 times as long as the other ==> other edge =6x

let height of box be y

volume of box v =x*6x*y

v=6x2y ==>y =v/6x2

given volume v is constant

surface area of open top box s=(x*6x)+(2*x*y)+(2*6x*y)

s=(6x2)+(14x)y

s=(6x2)+(14x)v/6x2

s=(6x2)+ 7v/(3x)

minimum surface area ==>ds/dx =0

ds/dx =(12x)- 7v/(3x2)=0

==>7v=36x3

==>x=(7v/36)1/3

one edge =(7v/36)1/3,other edge =6(7v/36)1/3==>(42v)1/3

y =v/6x2

y =v/(6((7v/36)1/3)2)

y=(6v/49)1/3

dimensions are (7v/36)1/3, (42v)1/3,(6v/49)1/3

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

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