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Sebiji cawan berbentuk silinder disediakan seperti dalam rajah.  


Cawan itu diisi penuh dengan air minuman dan hanya akan digunakan sekali. 

Berapakah bilangan biji cawan yang perlu disediakan? 

UNIT 7: ITEM 7c

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Air minuman yang disediakan tidak mencukupi untuk seorang sebiji cawan bagi 550 orang. 

Saiz botol air minuman yang baharu dengan ketinggian yang sama tetapi jejari ialah tiga kali botol air minuman asal telah digantikan. 

Adakah saiz botol air minuman yang baharu mencukupi.
Tandakan (✓) pada ruang yang disediakan. 

To solve the problem, we need to calculate and compare the volumes of the original and the new cylindrical cups.

### Step 1: Calculate the Volume of the Original Cup

The formula for the volume \( V \) of a cylinder is given by:

\[
V = \pi r^2 h
\]

Where:
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder.

From the image:
- The diameter of the original cup is \( 6 \, \text{cm} \), so the radius \( r \) is \( 3 \, \text{cm} \).
- The height \( h \) is \( 9 \, \text{cm} \).

Thus, the volume \( V_1 \) of the original cup is:

\[
V_1 = \pi \times (3 \, \text{cm})^2 \times 9 \, \text{cm} = 81\pi \, \text{cm}^3
\]

### Step 2: Calculate the Volume of the New Cup

The new cup has the same height but three times the radius of the original. So:
- New radius \( r_{\text{new}} = 3 \times 3 \, \text{cm} = 9 \, \text{cm} \).
- The height remains \( 9 \, \text{cm} \).

The volume \( V_2 \) of the new cup is:

\[
V_2 = \pi \times (9 \, \text{cm})^2 \times 9 \, \text{cm} = 729\pi \, \text{cm}^3
\]

### Step 3: Compare the Volumes

To determine whether the new bottle size is sufficient, we compare the volumes:

\[
\frac{V_2}{V_1} = \frac{729\pi \, \text{cm}^3}{81\pi \, \text{cm}^3} = \frac{729}{81} = 9
\]

The new cup holds 9 times the volume of the original cup.

### Step 4: Calculate Number of Original Cups for 550 People

Each person requires 1 original cup:

- Total volume needed for 550 people = \( 550 \times V_1 = 550 \times 81\pi \, \text{cm}^3 = 44550\pi \, \text{cm}^3 \).

### Step 5: Number of New Cups Required

Now, since each new cup holds 9 times the original:

\[
\text{Number of new cups required} = \frac{44550\pi \, \text{cm}^3}{729\pi \, \text{cm}^3} = \frac{44550}{729} \approx 61.11
\]

This calculation indicates that you would need approximately 62 new cups to provide the same volume of water.

### Conclusion

Given that the new cup holds 9 times the volume, significantly fewer cups are needed. For all 550 people, the volume provided by approximately 62 new cups is sufficient.

Would you like more details or have any questions?

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### Related Questions:
1. How would the result change if the height of the new bottle was doubled instead of the radius?
2. What would be the new volume if the radius and height were both tripled?
3. How much more volume does the new bottle provide compared to the original in terms of percentage?
4. What is the formula to determine the number of cups needed if the cup's radius is doubled?
5. What would be the impact on the number of cups needed if the number of people increased to 1000?

### Tip:
Always check the units and conversions when dealing with volume and area calculations to ensure accuracy.

Learn how to calculate the volume of a cylindrical cup and compare it to a new bottle size. This problem involves geometric calculations of cylinder volume, ratio, and proportion concepts. Suitable for students in Grades 7-9.

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