Let's tackle the two parts of your question one by one.
Part 1: Height of the Cone
Given a cone's vertical angle (the angle at the apex made by the two sides of the cone) is 70°, and the slant height (l) is 11 m.
To find the height (h) of the cone, we can use trigonometric relationships. The vertical angle is split into two right triangles. Each of these triangles has:
- The angle at the top: 70° / 2 = 35°.
- The slant height as the hypotenuse (l = 11 m).
Now we can find the height using the sine function: \[ \sin(35°) = \frac{h}{l} \] Rearranging this gives: \[ h = l \cdot \sin(35°) \]
Using a calculator to find \( \sin(35°) \): \[ \sin(35°) \approx 0.5736 \] Now substituting the values: \[ h = 11 \cdot 0.5736 \approx 6.31 \text{ m} \]
The height of the cone is approximately 6.31 meters.
Part 2: Depth of Water in the Cylindrical Tank
Given:
- Volume of water = 1760 liters (1 liter = 1000 cm³, so 1760 liters = 1760000 cm³)
- Diameter of the tank = 200 cm (thus, the radius \( r = \frac{200}{2} = 100 \) cm)
The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \] where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height (or depth of water in this case).
Substituting the values we know: \[ 1760000 = \frac{22}{7} \cdot (100)^2 \cdot h \] Calculating \( (100)^2 \): \[ (100)^2 = 10000 \] Now plugging it into the volume formula: \[ 1760000 = \frac{22}{7} \cdot 10000 \cdot h \]
Multiplying \( \frac{22}{7} \) by \( 10000 \): \[ \frac{22}{7} \cdot 10000 = \frac{220000}{7} \approx 31428.57 \]
Now rearranging for \( h \): \[ h = \frac{1760000 \cdot 7}{220000} = \frac{12320000}{220000} \approx 56 \]
Thus, the depth of water in the tank is approximately 56 cm.
Summary
- Height of the cone: 6.31 meters.
- Depth of water in the tank: 56 cm.
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