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Chapter 19: Problem 1857

Right circular cone has a height \(40 \mathrm{~cm}\) and its semi vertical angle is \(45^{\circ}\) then radius of its base circle is (a) \(40 \mathrm{~cm}\) (b) \(80 \mathrm{~cm}\) (c) \([(40 \sqrt{3}) / 2] \mathrm{cm}\) (d) \(20 \mathrm{~cm}\)

Short Answer

Expert verified
The radius of the base circle of the right circular cone is \(40 \mathrm{~cm}\), which corresponds to option (a) in the exercise.

Step by step solution

01

Use the tangent function relation

We know that the tangent function is defined as: \[\tan (\angle) = \frac{opposite}{adjacent}\] In our case, the semi-vertical angle is given as 45°, the height of the cone is the adjacent side (40 cm), and the radius of the base is the opposite side. So we have: \[\tan (45°) = \frac{radius}{40 \mathrm{~cm}}\] We need to calculate the radius from this equation.
02

Calculate the value of the tangent function for the given angle

The tangent of 45° is 1. So we have: \[\tan (45°) = 1 \] Substitute this value back into our equation: \[1 = \frac{radius}{40 \mathrm{~cm}}\]
03

Solve for the radius

Now, we can solve for the radius: \[radius = 1 \times 40 \mathrm{~cm}\] \[radius = 40 \mathrm{~cm}\]
04

Conclusion

The radius of the base circle of the right circular cone is 40 cm, which corresponds to option (a) in the exercise.

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