A rain drop hitting a lake makes a circular ripple If the radius – Free 83A

A rain drop hitting a lake makes a circular ripple. If the radius, in inches, grows as a function of time in minutes according to r(t) = 25 ?— t + 2?, fi d the area of the ripple as a function of time. Find the area of the ripple at t = 2.

Answer

Understanding Ripple Area Growth Over Time with r(t) = 25√t + 2

When a raindrop hits a still lake, it creates a mesmerizing circular ripple that expands outward. In this fascinating scenario, the radius of the ripple is not constant—it changes over time. Let’s analyze how to find the area of the ripple as a function of time, and then calculate the exact area at a specific moment: t = 2 minutes.

Given Function:

The radius as a function of time is defined as:

r(t) = 25√t + 2

Where:

• t is time in minutes
• r(t) is the radius of the ripple in inches

Step 1: Formula for the Area of a Circle

The general formula to find the area A of a circle is:

A = π × [r(t)]²

Step 2: Substitute r(t) into the Area Formula

We substitute the radius function into the area formula:

A(t) = π × (25√t + 2)²

Step 3: Expand the Binomial

Now we expand (25√t + 2)² using the identity (a + b)² = a² + 2ab + b²:

• a = 25√t, b = 2
• a² = (25√t)² = 625t
• 2ab = 2 × 25√t × 2 = 100√t
• b² = 2² = 4

So,

A(t) = π × (625t + 100√t + 4)

Step 4: Final Expression for the Area

The area of the ripple as a function of time is:

A(t) = π(625t + 100√t + 4) square inches

Step 5: Find the Area at t = 2

Now, substitute t = 2 into the function:

• 625 × 2 = 1250
• 100√2 ≈ 100 × 1.4142 = 141.42
• Constant = 4

So the total expression becomes:

A(2) = π × (1250 + 141.42 + 4) = π × 1395.42

Now approximate using π ≈ 3.1416:

A(2) ≈ 3.1416 × 1395.42 ≈ 4382.55 square inches

Conclusion:

The area of the circular ripple created by the raindrop after 2 minutes is approximately:

4382.55 square inches