A cone of height 9cm has a volume of n cm3 and a curved surface area of n cm2. Find the vertical angle of the cone
ncm³=ncm²
⅓πr²h=πrl
⅓πr²9=πrl
⅓πr²9/πr=πrl/πr
9r/3=l
Cross multiply
9r=3l
r=3l/9
r=⅓l
Divide both sides by l
r/l=⅓
r/l=0.3333=sin°
Sin-1°=19.471
Vertical angle=2(°)
2(°)=2(19.471)
2(°)=38.94
…Vertical angle=38.9(to 0.1°)
let the radius be r and the slant height be L
then first of all: L^2 = r^2 + 9^2
= r^2 + 81
Surface area = πrL = n
volume = (1/3)πr^2 (9) = 3πr^2 = n
then 3πr^2 = πrL
3r = L , then in L^2 = r^2 + 81
9r^2 = r^2 + 81
8r^2 = 81
r = 9/√8
L^2 = 81/8 + 81 = 729/8
L = √1458/4
cos(baseangle) = (9/√8) / (√1458/4)
= 1/3
angle of base = appr 70.5°
The answer is 38.9.
I don't understand how you did it.
I agree, I got the same
I don't understand how you did it
I don't understand how you did it
ino get
how was the square root of 1458/4 gotten
To find the vertical angle of the cone, we can use the formula for the slant height and the height of the cone.
The formula for the slant height (l) of a cone is given by:
l = √(h^2 + r^2)
where h is the height of the cone and r is the radius of the base.
Given that the height of the cone is 9 cm, we need to find the radius.
The formula for the volume of a cone (V) is given by:
V = (1/3)πr^2h
We are given that the volume is n cm^3. Substituting the values into the formula, we have:
n = (1/3)πr^2(9)
Simplifying the equation, we get:
3n = 9πr^2
Dividing both sides by 9π, we get:
r^2 = (3n)/(9π)
Taking the square root of both sides, we have:
r = √[(3n)/(9π)]
Now, substituting the value of r into the formula for the slant height, we get:
l = √(9^2 + [(3n)/(9π)]^2)
Simplifying further:
l = √(81 + (9n^2)/(81π^2))
Next, we use the formula for the curved surface area (A) of a cone, which is given by:
A = πrl
We are given that the curved surface area is n cm^2. Substituting the values into the formula, we have:
n = πr(√(81 + (9n^2)/(81π^2)))
Simplifying the equation, we get:
n = √(81n^2 + 9n^2/(π^2))
Squaring both sides to eliminate the square root, we have:
n^2 = 81n^2 + 9n^2/(π^2)
Combining like terms:
0 = 81n^2 + 9n^2/(π^2) - n^2
0 = n^2(81 + 9/(π^2) - 1)
0 = n^2(81 + 9/(π^2) - 1)
Simplifying further, we get:
0 = n^2(80 + 9/(π^2))
Since n^2 cannot be zero, we focus on the expression in the parentheses:
80 + 9/(π^2) = 0
Solving for π, we have:
π^2 = -9/80
However, this yields a negative value for π^2, which is not possible. Therefore, there is no real solution for the vertical angle of the cone given the information provided.