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 [tex]\(\sqrt{2}\)[/tex].
Let [tex]\( r \)[/tex] be the radius of the cone and [tex]\( h \)[/tex] be the height of the cone. Since it’s a 45-45-90 triangle:
[tex]\[ \text{Slant Height} = r \sqrt{2} \][/tex]
[tex]\[ 12 = r \sqrt{2} \][/tex]
Step 3: Solving for the radius [tex]\( r \)[/tex]
Isolating [tex]\( r \)[/tex]:
[tex]\[ r = \frac{12}{\sqrt{2}} \][/tex]
This simplifies to:
[tex]\[ r = \frac{12}{\sqrt{2}} = 12 \div \sqrt{2} \][/tex]
To rationalize the denominator:
[tex]\[ r = 12 \div \sqrt{2} \times \frac{\sqrt{2}}{\sqrt{2}} = 12 \times \frac{\sqrt{2}}{2} = 6\sqrt{2} \][/tex]
[tex]\[ r = 6\sqrt{2} \approx 8.485 \text{ units} \][/tex]
So, the radius of the cone is approximately [tex]\( 8.485 \)[/tex] units.
Step 4: Finding the diameter
The diameter [tex]\( d \)[/tex] of the cone is twice the radius:
[tex]\[ d = 2 \times r \][/tex]
[tex]\[ d = 2 \times 8.485 \approx 16.971 \text{ units} \][/tex]
So, the diameter of the cone is approximately [tex]\( 16.971 \)[/tex] units.
Step 5: Finding the height
In a 45-45-90 triangle, the height [tex]\( h \)[/tex] is equal to the radius:
[tex]\[ h = r = 8.485 \text{ units} \][/tex]
Summary:
a) The diameter of the cone is approximately [tex]\( 16.971 \)[/tex] units.
b) The height of the cone is approximately [tex]\( 8.485 \)[/tex] units.
