3.1 Q-5

Question Statement

Find the approximate increase in the area of a circular disc if its diameter is increased from 44 cm to 44.4 cm.

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Background and Explanation

To solve this problem, we use the formula for the area of a circle:
$A = \pi r^2$
where $r$ is the radius of the circle.
The radius is half the diameter, so any change in diameter directly affects the radius. The goal is to approximate the increase in area using the concept of differentials:
$\mathrm{d}A = \frac{\mathrm{d}A}{\mathrm{d}r} \cdot \mathrm{d}r$
This approach avoids recalculating the area from scratch by leveraging the small change ($\mathrm{d}r$) in radius.

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Solution

Step 1: Identify the Initial Radius and Change in Radius

• The initial diameter of the disc is $44 , \text{cm}$.
• The initial radius is:
  $r = \frac{\text{diameter}}{2} = \frac{44}{2} = 22 , \text{cm}$
• When the diameter increases to $44.4 , \text{cm}$, the new radius becomes:
  $r + \mathrm{d}r = \frac{44.4}{2} = 22.2 , \text{cm}$
• The change in radius is:
  $\mathrm{d}r = 22.2 - 22 = 0.2 , \text{cm}$

Step 2: Differentiate the Area Formula

The formula for the area of a circle is $A = \pi r^2$. Differentiating with respect to $r$:
$\frac{\mathrm{d}A}{\mathrm{d}r} = 2\pi r$

Step 3: Apply the Differential Formula

Using the differential approximation formula $\mathrm{d}A = \frac{\mathrm{d}A}{\mathrm{d}r} \cdot \mathrm{d}r$, substitute:
$\mathrm{d}A = 2\pi r \cdot \mathrm{d}r$

Step 4: Substitute the Known Values

Substitute $r = 22 , \text{cm}$, $\mathrm{d}r = 0.2 , \text{cm}$, and $\pi = 3.14$:
$\mathrm{d}A = 2 \cdot 3.14 \cdot 22 \cdot 0.2$
Simplify step by step:

1. Multiply $2 \cdot 3.14 = 6.28$.
2. Multiply $6.28 \cdot 22 = 138.16$.
3. Multiply $138.16 \cdot 0.2 = 27.632 , \text{cm}^2$.

Thus, the approximate increase in the area is:
$\mathrm{d}A \approx 27.63 , \text{cm}^2$.

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Key Formulas or Methods Used

1. Area of a Circle:
   $A = \pi r^2$
2. Differential Formula for Area:
   $\mathrm{d}A = \frac{\mathrm{d}A}{\mathrm{d}r} \cdot \mathrm{d}r$
3. Derivative of Area:
   $\frac{\mathrm{d}A}{\mathrm{d}r} = 2\pi r$

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Summary of Steps

1. Calculate the initial radius $r$ and the change in radius $\mathrm{d}r$.
2. Differentiate the area formula $A = \pi r^2$ to find $\frac{\mathrm{d}A}{\mathrm{d}r} = 2\pi r$.
3. Use $\mathrm{d}A = 2\pi r \cdot \mathrm{d}r$ to approximate the change in area.
4. Substitute the known values ($r = 22$, $\mathrm{d}r = 0.2$, $\pi = 3.14$) and simplify to find $\mathrm{d}A$.
5. The final result is $\mathrm{d}A \approx 27.63 , \text{cm}^2$.