Math Problem Statement

just answers please

A circle is growing, its radius increasing by 5 mm per second. Find the rate at which the area is changing at the moment when the radius is 22 mm.

Solution

To find the rate at which the area of the circle is changing (dAdt\frac{dA}{dt}) when the radius is 22 mm:

Formula:

The area of a circle is A=πr2A = \pi r^2.
Differentiate with respect to tt: dAdt=2πrdrdt\frac{dA}{dt} = 2\pi r \frac{dr}{dt}

Known values:

  • r=22mmr = 22 \, \text{mm}
  • drdt=5mm/s\frac{dr}{dt} = 5 \, \text{mm/s}

Calculation:

dAdt=2π(22)(5)\frac{dA}{dt} = 2\pi (22)(5) dAdt=220πmm2/s\frac{dA}{dt} = 220\pi \, \text{mm}^2/\text{s} Using π3.14159\pi \approx 3.14159: dAdt2203.14159=691.15mm2/s\frac{dA}{dt} \approx 220 \cdot 3.14159 = 691.15 \, \text{mm}^2/\text{s}

Final Answer:

691.150mm2/s\boxed{691.150 \, \text{mm}^2/\text{s}}

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Related Rates

Formulas

Area of a circle: A = πr²
Rate of change formula: dA/dt = 2πr(dr/dt)

Theorems

Derivative rules
Rate of change applications

Suitable Grade Level

Grade 11-12 or introductory college-level calculus