let p(x) = 4x+5 Represent the radial density of a disk. Calculate the mass of a disk within 3 cm. Round to 4 decimal places 

To calculate the mass of the disk within a radius of 3 cm, we can use the following steps:

1. **Mass formula for a disk**:  
   The mass \( M \) of a disk with variable radial density is given by the formula:
   \[
   M = \int_0^R \rho(r) \cdot 2\pi r \, dr
   \]
   where:
   - \( \rho(r) \) is the radial density function (in this case, \( p(x) = 4r + 5 \)).
   - \( R \) is the radius of the disk (in this case, 3 cm).
   - \( 2\pi r \) accounts for the circumference at a given radius \( r \).

2. **Substitute the density function \( \rho(r) = 4r + 5 \) into the mass formula**:
   \[
   M = \int_0^3 (4r + 5) \cdot 2\pi r \, dr
   \]

3. **Solve the integral**:
   First, expand the expression:
   \[
   M = 2\pi \int_0^3 (4r^2 + 5r) \, dr
   \]
   Now, compute the integral:
   \[
   M = 2\pi \left[ \frac{4r^3}{3} + \frac{5r^2}{2} \right]_0^3
   \]
   Substituting \( r = 3 \) into the expression:
   \[
   M = 2\pi \left[ \frac{4(3)^3}{3} + \frac{5(3)^2}{2} \right]
   \]
   Simplify:
   \[
   M = 2\pi \left[ \frac{4(27)}{3} + \frac{5(9)}{2} \right]
   \]
   \[
   M = 2\pi \left[ 36 + 22.5 \right]
   \]
   \[
   M = 2\pi (58.5)
   \]
   \[
   M = 117\pi
   \]

4. **Approximate the result**:
   \[
   M \approx 117 \times 3.1416 = 367.5667
   \]

Thus, the mass of the disk is approximately **367.5667 grams** (rounded to four decimal places).

Would you like further details or have any questions?

Here are five related questions you might find useful:
1. How does the density function affect the mass of the disk?
2. What happens if the radius is increased to 5 cm?
3. Can you explain why \( 2\pi r \) is used in the formula?
4. What would the mass be if the density was constant (e.g., \( \rho(r) = 5 \))?
5. How would the mass change if the density function was quadratic, say \( p(x) = r^2 + 4r + 5 \)?

**Tip**: When dealing with circular objects, always ensure you include \( 2\pi r \) when integrating over the radius for mass or area-based quantities!

let p(x) = 4x + 5 represent the radial density of a disk. Calculate the mass of a disk within 3 cm. Round to 4 decimal places.

Learn how to calculate the mass of a disk with a radial density function p(x) = 4x + 5, using integral calculus. This example covers key concepts in calculus and physics, including the use of the mass integral for a disk of radius 3 cm. Suitable for advanced high school students (Grades 11-12).

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