https://r2.erweima.ai/mathbot/image/1726498427612txrg4etj.jpg

Let's solve the given problems based on the cone-shaped party hat.

### Given:
- Height of the cone \( h = 20 \, \text{cm} \)
- Radius of the base \( r = 7 \, \text{cm} \)
- Slant height (generatrix) \( g = 21 \, \text{cm} \)

#### Problem 7.1: 
Determine the length of the arc \( AB \).

The arc length \( AB \) is the circumference of the base of the cone scaled down by the fraction of the full circle used in the cone's surface.

The full circumference of a circle with radius \( g \) (slant height) is:
\[
C = 2\pi g = 2\pi \times 21 = 42\pi \, \text{cm} \approx 131.95 \, \text{cm}
\]

Now, the angle \( \theta \) of the sector can be calculated using the ratio of the base circumference to the full circumference:
\[
\theta = \frac{\text{circumference of the base}}{\text{circumference of the full circle}} \times 360^\circ
\]
\[
\text{Circumference of the base} = 2\pi r = 2\pi \times 7 = 14\pi \, \text{cm} \approx 43.98 \, \text{cm}
\]

Now, we calculate the angle:
\[
\theta = \frac{14\pi}{42\pi} \times 360^\circ = \frac{1}{3} \times 360^\circ = 120^\circ
\]

Thus, the arc length \( AB \) is given by:
\[
\text{Arc length} = \frac{120^\circ}{360^\circ} \times 42\pi = \frac{1}{3} \times 42\pi \approx 43.98 \, \text{cm}
\]

So, the arc length is approximately **44 cm**.

#### Problem 7.2: 
Determine the approximate area of the cardboard used in constructing the hat.

The area of the sector forming the cone's lateral surface is given by:
\[
A = \frac{\theta}{360^\circ} \times \pi g^2
\]
\[
A = \frac{120^\circ}{360^\circ} \times \pi \times 21^2 = \frac{1}{3} \times \pi \times 441
\]
\[
A \approx \frac{1}{3} \times 1385.44 \, \text{cm}^2 \approx 461.81 \, \text{cm}^2
\]

Thus, the area of the cardboard used is approximately **461.8 cm²**.

### Do you want more details or have any other questions? Here are 5 related questions to deepen your understanding:
1. How would the area of the cardboard change if the slant height increased to 25 cm?
2. What would the arc length be if the radius of the base was increased to 10 cm?
3. Can you determine the surface area of the base of the cone?
4. How do you calculate the total surface area of a cone including the base?
5. If the height of the cone was doubled, how would the angle of the sector change?

### Tip:
For problems involving cones, the slant height plays a crucial role in determining both the lateral surface area and the arc length in its net.

Determine the length of the arc AB and the area of the cardboard used in constructing a cone-shaped party hat. Given: height = 20 cm, base radius = 7 cm, slant height = 21 cm.

Learn how to calculate the arc length and the surface area of the cardboard used to construct a cone-shaped party hat. This problem covers geometric concepts such as the circumference of a circle, the area of a sector, and the use of slant height in cones. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation