Math Is Fun Forum

Hastega

Member

Registered: 2007-06-20

Posts: 7

Please help me, Differential Calculus

My teacher gave us an assignment he never discussed with my class, please help me with this Differential Calculus problem.

1.) The product of two positive numbers is 100, If the sum of their cubes is a minimum, find the numbers.

2.) A poster is to have an area of 120 square inches with one inch margins at the bottom and sides and  2 inch margin at the top. What dimensions will give the largest printed area?

3.)An open box with a square base is to contain 100 cm^3. Find it's dimensions for a minimum amount of material in making the box.

4.) A ladder 13 feet long rests againts a vertical wall. If the bottom of the ladder slides away from the wall at the rate of 0.5 ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 feet from the wall?

5.) At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 20 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when it is 10 feet high?

6.) The radius of a spherical ballon is increasing at a rate of 5 inches per minute. Find the rate of change of the volume when
r = 3 inches.

7.) Two cars start moving from the same point. One travels south at 40 m/hr and the other travels at 30 m/hr. At what rate is the distance between the cars increasing two hours later?

Help is greatly appreciated.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Please help me, Differential Calculus

1)
We have two numbers x and y such that xy = 100. Therefore, y = 100/x.
We want to minimise x³ + y³. This is the same as x³ + (100/x)³.

Any minimum points that this function has will be at points where the derivative is 0.
The derivative is 3x² - 3000000x[sup]-4[/sup].

Equating to 0 and simplifying gives that x[sup]6[/sup] = 1000000.
We are told that x is positive and so this means that x = 10.

That means that y is also 10 and the sum of their cubes is 2000.

2)
If the poster's width is x inches, then its height must be (120/x) inches, for the area to be correct.
The printed width is 2 less than the actual width, because there is one inch taken off each side, and the printed height is 3 less than the actual height.

Therefore, we want to maximise the following function: (x-2)[(120/x)-3]
Expanding: 120 - 3x - 240/x + 6
Differentiate and equate to 0: -3 + 240/x² = 0
Solve: 240 = 3x²
x² = 80
x = √80 ≈ 8.94 inches.

This gives a printed area of ≈ 72.3 square inches.

The rest of the questions are dealt with in a similar way. Just turn them into an equation (or set of equations) and solve them from there.

---

Why did the vector cross the road?
It wanted to be normal.

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Please help me, Differential Calculus

6.) The radius of a spherical ballon is increasing at a rate of 5 inches per minute. Find the rate of change of the volume when
r = 3 inches.

r = 3

---

The Beginning Of All Things To End.
The End Of All Things To Come.

Hastega

Member

Registered: 2007-06-20

Posts: 7

Re: Please help me, Differential Calculus

@Math & Luca: Thanks alot guys, I only need to do 4,5 and 7 because I solved 3 today.