High School

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19.1 Determine \( A \), the area indicated on the top layer of the chocolate cake in square millimeters (mm²). Give your answer to the nearest whole millimeter.

19.2 A ribbon will be wrapped around the circumference of the bottom layer of the cake and must overlap by 50 mm. Calculate the length of the ribbon in mm. You may use the formula: Circumference = \( 2 \times \pi \times \text{radius} \) where \( \pi = 3.142 \).

19.3 The volume of the rectangular cardboard box is 42,000 cm³. Determine the height of the box in cm.

19.4 Mr. Johnathan's cousin in England claims that the diameter of the bottom layer cake is more than 10 inches. Verify with the necessary calculation if his claim is valid or not.

**NOTE:**
- Diameter of cake (top layer): 18 cm
- Diameter of cake (bottom layer): 250 mm
- Area of a circle = \( \pi \times (\text{radius})^2 \)
- Volume of a rectangular box = \( l \times w \times h \) where \( l = \text{length}; w = \text{width}; h = \text{height} \)
- 1 inch = 25.4 mm

Answer :

Let's go through each part of the problem step by step:

19.1 Calculate the Area of the Top Layer of the Cake

  • Given that the diameter of the top layer of the cake is 18 cm, we first convert it to millimeters: 18 cm = 180 mm.
  • The radius [tex]r[/tex] of the top layer is half of the diameter: [tex]r = \frac{180}{2} = 90[/tex] mm.
  • Using the formula for the area of a circle, [tex]A = \pi r^2[/tex], where [tex]\pi = 3.142[/tex]:
    [tex]A = 3.142 \times (90)^2[/tex]
    [tex]A = 3.142 \times 8100[/tex]
    [tex]A \approx 25434.2 \text{ mm}^2[/tex]
  • Rounded to the nearest whole millimeter, the area is approximately 25,434 mm².

19.2 Calculate the Length of the Ribbon

  • The diameter of the bottom layer is provided as 250 mm, thus the radius [tex]r = \frac{250}{2} = 125[/tex] mm.
  • The formula for the circumference is [tex]C = 2\pi r[/tex]:
    [tex]C = 2 \times 3.142 \times 125[/tex]
    [tex]C = 785.5 \text{ mm}[/tex]
  • Since the ribbon must overlap by 50 mm, the total length of the ribbon is:
    [tex]785.5 + 50 = 835.5 \text{ mm}[/tex]
  • Therefore, the length of the ribbon is 836 mm (rounded to the nearest mm).

19.3 Determine the Height of the Box

  • We need to find the height [tex]h[/tex] of a rectangular box with a volume of 42,000 cm³, using the formula: [tex]V = l \times w \times h[/tex].
  • Without the values for length [tex]l[/tex] and width [tex]w[/tex], we cannot determine [tex]h[/tex] precisely.
  • Therefore, more information is needed to solve for the height of the box.

19.4 Verify the Claim About the Diameter of the Bottom Layer Cake

  • To convert 250 mm to inches, use the conversion 1 inch = 25.4 mm:
    [tex]\text{Diameter in inches} = \frac{250}{25.4}[/tex]
    [tex]\text{Diameter in inches} \approx 9.84[/tex]
  • Since 9.84 inches is less than 10 inches, Mr. Jonathan's cousin's claim that the diameter is more than 10 inches is not valid.

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Rewritten by : Barada