Math Problem Statement
Solution
I see the image you uploaded. Let's go over the problem step-by-step:
Problem 1:
The container has a height of 35 cm and a cross-section designed from three semicircles with diameters of 17.5 cm, 6.5 cm, and 24 cm. Here are the tasks:
(a) Calculate the area of the cross-section of the container.
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We need to find the areas of the semicircles and sum them up.
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The area of a semicircle is given by , where is the radius.
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For the first semicircle with a diameter of 17.5 cm: Area:
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For the second semicircle with a diameter of 6.5 cm: Area:
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For the third semicircle with a diameter of 24 cm: Area:
The total area will be:
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(b) Calculate the external surface area of the container, including the base.
- The surface area of a cylindrical container (without the top) can be calculated as: Where the circumference is the perimeter of the cross-section (sum of the semicircle circumferences).
(c) Calculate the capacity of the container.
The capacity (volume) of the container is given by: The height is 35 cm, and the cross-sectional area was calculated in part (a). Convert the volume from cubic centimeters to liters (1 cm³ = 0.001 liters).
(d) Pouring the water into a cone-shaped container.
The cone has a radius of 20 cm and a height of 40 cm. The volume of a cone is given by: We will check if the water from the original container can fill this cone.
I will begin calculating each part now.The area of the cross-section of the container is approximately 363.05 cm².
Would you like me to continue with part (b), or do you have any questions on this so far?
Here are some related questions to deepen your understanding:
- How would the calculation change if all three shapes were full circles instead of semicircles?
- Can you think of how the shape of the container affects its overall surface area?
- What impact does increasing the height of the container have on its capacity?
- How could the shape of the base influence the structural stability of the container?
- How would the material used for the container impact its external surface area?
Tip: Always double-check unit conversions when switching between cm² and liters!
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Math Problem Analysis
Mathematical Concepts
Geometry
Area Calculation
Surface Area
Volume Calculation
Unit Conversion
Formulas
Area of semicircle: A = 0.5 * π * r^2
Circumference of semicircle: C = π * r
Surface area of a cylinder: SA = circumference * height + base area
Volume of a cylinder: V = base area * height
Volume of a cone: V = (1/3) * π * r^2 * h
Unit conversion: 1 cm³ = 0.001 liters
Theorems
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Suitable Grade Level
Grades 9-12
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