A toy is in the form of a cone mounted on a hemisphere of radius 7 cm. The total height of the toy is 31 cm. Find the total surface area of the toy.
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Always factor out common terms like \( \pi r \) before substituting values.
This reduces the number of multiplications and divisions, saving time and minimizing calculation errors.
Recognizing standard Pythagorean triples like (7, 24, 25) instantly gives the slant height without doing square root calculations.
Step 1: Understanding the Question:
The topic of this question is Surface Areas and Volumes, specifically dealing with a combination of solids.
The toy consists of two parts: a cone at the top and a hemisphere at the bottom.
The base of the cone is mounted flush on the flat circular face of the hemisphere, meaning both share the same radius.
To find the total surface area of the toy, we need to add the curved surface area (CSA) of the cone and the curved surface area (CSA) of the hemisphere.
Step 2: Key Formula or Approach:
- Radius of the hemisphere and the cone, \( r = 7\text{ cm} \).
- Curved Surface Area of a hemisphere:
\[ \text{CSA}_{\text{hemisphere}} = 2\pi r^2 \]
- Curved Surface Area of a cone:
\[ \text{CSA}_{\text{cone}} = \pi r l \]
Where \( l \) is the slant height of the cone, given by:
\[ l = \sqrt{r^2 + h^2} \]
- Here, \( h \) is the height of the conical part of the toy.
Step 3: Detailed Explanation:
1. First, find the height of the conical part \( h \):
The total height of the toy is 31 cm.
This total height is the sum of the height of the cone \( h \) and the radius of the hemisphere \( r \):
\[ \text{Total Height} = h + r \]
\[ 31 = h + 7 \]
\[ h = 31 - 7 = 24\text{ cm} \]
2. Calculate the slant height \( l \) of the cone:
Using the relation \( l = \sqrt{r^2 + h^2} \):
\[ l = \sqrt{7^2 + 24^2} \]
\[ l = \sqrt{49 + 576} \]
\[ l = \sqrt{625} \]
\[ l = 25\text{ cm} \]
3. Write the formula for the total surface area (TSA) of the toy:
\[ \text{TSA} = \text{CSA}_{\text{cone}} + \text{CSA}_{\text{hemisphere}} \]
\[ \text{TSA} = \pi r l + 2\pi r^2 \]
Factor out common terms to make the calculation simpler:
\[ \text{TSA} = \pi r (l + 2r) \]
4. Substitute the known values into the simplified expression:
Take \( \pi = \frac{22}{7} \), \( r = 7\text{ cm} \), and \( l = 25\text{ cm} \):
\[ \text{TSA} = \frac{22}{7} \times 7 \times (25 + 2 \times 7) \]
\[ \text{TSA} = 22 \times (25 + 14) \]
\[ \text{TSA} = 22 \times 39 \]
5. Perform the final multiplication:
\[ \text{TSA} = 858\text{ cm}^2 \]
Step 4: Final Answer:
The total surface area of the toy is \(858\text{ cm}^2\).
