# sand is dumped such that the shape of the sandpile

questioned by administrator @ in Mathematics viewed by 286 People

5. Sand is dumped off a conveyor belt into a pile at the rate of 2 cubic feet per minute. The sand pile is shaped like a cone whose height and base diameter are always equal. At what rate is the height of the pile growing when the pile is 5 feet high? (The volume of a cone is fr2h where r is the radius of the base and h is the height:)​

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
7. Subtraction
• Left to Right

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Subtract Property of Equality

Geometry

Volume of a Cone:

• h is height

Diameter: d = 2r

Calculus

Derivatives

Derivative Notation

Differentiating with respect to time

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Step-by-step explanation:

Step 1: Define

Step 2: Rewrite Volume Formula

We need to rewrite the cone volume formula in terms of height h only.

Base b = diameter d of the circular base

1. Define:                    b = d
2. Substitute:              b = 2r

We are given that the base of the cone is the same as the height.

1. Define:                    b = 2r
2. Substitute:              h = 2r

Now solve for height.

1. Divide 2 on both sides:                    h/2 = r
2. Rewrite expression:                          r = h/2

Now find new volume formula.

1. Define [VC]:
2. Substitute:
3. Exponents:
4. Multiply:

We now have the same volume formula in terms of height h only.

Step 3: Differentiate

1. Basic Power Rule:
2. Simplify:

Step 4: Solve for height rate

1. Substitute:
2. Isolate h rate:
3. Exponents:
4. Multiply:
5. Divide:
6. Rewrite:

Here this tells us that the rate at which the height is moving at a rate of 0.101859 feet per minute.

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