To solve the problem of finding the ratio of the lateral area of a cone to the lateral area of a cylinder when they have equal radii [tex]\(r\)[/tex] and equal heights [tex]\(h\)[/tex], let’s break it down step by step.
### 1. Calculate the Slant Height of the Cone
The slant height [tex]\(l\)[/tex] of the cone can be determined using the Pythagorean theorem:
[tex]\[ l = \sqrt{r^2 + h^2} \][/tex]
### 2. Calculate the Lateral Area of the Cone
The formula for the lateral area of the cone is given by:
[tex]\[ \text{Lateral Area of Cone} = \pi r l \][/tex]
Substituting [tex]\(l\)[/tex] from the previous step:
[tex]\[ \text{Lateral Area of Cone} = \pi r \sqrt{r^2 + h^2} \][/tex]
### 3. Calculate the Lateral Area of the Cylinder
The lateral area of the cylinder is calculated by:
[tex]\[ \text{Lateral Area of Cylinder} = 2 \pi r h \][/tex]
### 4. Find the Ratio of the Lateral Areas
Now, we need to find the ratio of the lateral area of the cone to the lateral area of the cylinder:
[tex]\[ \text{Ratio} = \frac{\text{Lateral Area of Cone}}{\text{Lateral Area of Cylinder}} \][/tex]
Substituting the formulas:
[tex]\[ \text{Ratio} = \frac{\pi r \sqrt{r^2 + h^2}}{2 \pi r h} \][/tex]
Simplifying:
[tex]\[ \text{Ratio} = \frac{\sqrt{r^2 + h^2}}{2h} \][/tex]
Given the information from the result:
- Lateral Area of the Cone = 4.442882938158366
- Lateral Area of the Cylinder = 6.283185307179586
- Ratio = 0.7071067811865476
The ratio of the lateral area of the cone to the lateral area of the cylinder is approximately [tex]\( \frac{1}{\sqrt{2}} \)[/tex], or in a more simplified numerical form:
[tex]\[ \text{Ratio} \approx 0.707 \][/tex]
### Final Answer
From the given options, the correct choice that reflects the ratio derived is:
[tex]\[ \boxed{\frac{1}{2 \sqrt{r^2 + h^2}}: 1} \][/tex] (though this specific ratio may not be listed among the options directly, it indicates the correct comparative form of the areas). If the options simplify differently in wording, verify the exact match of your detailed derived ratio.
