Certainly! Let's solve the problem step-by-step.
Given:
- The vertex angle of the cone is 90°.
- The slant height of the cone is 12 units.
Step 1: Understanding the triangle
Since the vertex angle is 90°, the triangle formed by the slant height, radius, and height of the cone is a right-angled triangle with angles 45°, 45°, and 90°. This is known as a 45-45-90 triangle.
Step 2: Characteristics of a 45-45-90 triangle
In a 45-45-90 triangle:
- The two legs are equal.
- The hypotenuse (which is the slant height in our case) is the length of one leg multiplied by \(\(\sqrt{2}\)\).
Let \(\( r \)\) be the radius of the cone and \(\( h \)\) be the height of the cone. Since it’s a 45-45-90 triangle:
\(\[ \text{Slant Height} = r \sqrt{2} \]\)
\(\[ 12 = r \sqrt{2} \]\)
Step 3: Solving for the radius \(\( r \)\)
Isolating \(\( r \)\):
\(\[ r = \frac{12}{\sqrt{2}} \]\)
This simplifies to:
\(\[ r = \frac{12}{\sqrt{2}} = 12 \div \sqrt{2} \]\)
To rationalize the denominator:
\(\[ r = 12 \div \sqrt{2} \times \frac{\sqrt{2}}{\sqrt{2}} = 12 \times \frac{\sqrt{2}}{2} = 6\sqrt{2} \]\)
\(\[ r = 6\sqrt{2} \approx 8.485 \text{ units} \]\)
So, the radius of the cone is approximately \(\( 8.485 \)\) units.
Step 4: Finding the diameter
The diameter \(\( d \)\) of the cone is twice the radius:
\(\[ d = 2 \times r \]\)
\(\[ d = 2 \times 8.485 \approx 16.971 \text{ units} \]\)
So, the diameter of the cone is approximately \(\( 16.971 \)\) units.
Step 5: Finding the height
In a 45-45-90 triangle, the height \(\( h \)\) is equal to the radius:
\(\[ h = r = 8.485 \text{ units} \]\)
Summary:
a) The diameter of the cone is approximately \(\( 16.971 \)\) units.
b) The height of the cone is approximately \(\( 8.485 \)\) units.
