4 geometry questions?

Question

6.
What is the measure of one edge of a cube if the diagonal of one of its faces is 4√3 units?

15.
What is the slant height of a right circular cone whose lateral surface area is 66pie units^2 and whose radius is 6 units?

18.
The total area of a cone is 38pie meters^2 and its radius is 2 meters. Find the slant height.

20.
A birthday hat in the shape of a cone has a lateral surface of 100pie cm^2. The slant height is 25 cm. Find the radius of the hat.


submission 31

Answer 1

Q.6.
What is the measure of one edge of a cube if the diagonal of one of its faces is 4√3 units?
Solution:
We know the formula, if "a" is the side/edge of a cube then its diagonal will be a√3.
Therefore 4√3 =a√3 or a =4 units......................i
Q.15.
What is the slant height of a right circular cone whose lateral surface area is 66pie units^2 and whose radius is 6 units?
Ans: If "r" is the radius and "h" is the height of the cone and "l" is the slant height then
Lateral surface area =pie r l=pie(6)(l)=66 pie
or l=11 units...............................................ii

Q.18.
The total area of a cone is 38pie meters^2 and its radius is 2 meters. Find the slant height.
Ans:
Use the above formula
Area of cone =1/3pie r^2 h=38 pi
1/3 pie(2)^2(h)=38 pie
h=38(3)/4=28.5 unit
l^2= h^2+r^2
l^2= 28.5^2+2^2=816.25
l =28.57 units................................iii



Q.20.
A birthday hat in the shape of a cone has a lateral surface of 100pie cm^2. The slant height is 25 cm. Find the radius of the hat.
Ans: Using above formula
Lateral surface =pie r (25)=pie 100
r=4 units..............................iv

Answer 2

For the first problem, you know that a square (one face of a cube) is made up of right angles. Therefore you will use the 45-45-90 triangle rule. The two legs are x and the diagonal part (half of a square is a triangle) is x√2. This means that 4√3 needs to be put in a form so that we can determine x. 4√3 is also equal to √48 because √(16 times 3). Thus, we can also make √48 equal to √24 x √2. This makes √24 equal to x. √24 is also the length of one edge of the cube. You can also simplify √24 into √(6 x 4) which becomes 2√6.

Lateral Area = (pie • radius • slant height)
You can also rearrange this equation so that:
Slant height= Lateral Area/ (pie • radius)

For the second problem you simply plug in the givens and solve for slant height.

Slant height=66 pie units^2 / (pie • 6 units)
=(66 pie units) / (6 pie)
=11 units = slant height.

For the third problem you solve the problem in a similar manner.
Slant height = (38 pie meters^2)/(pie • 2 meters)
= (38 pie meters)/(2 pie)
= 19 meters = slant height.

The last problem uses the same equation. By rearranging the equation once again you get:
Radius = Lateral area/ (pie • slant height)

Radius = (100 pie cm^2)/ (pie • 25 cm)
= (100 pie cm)/ (pie • 25)
= (100 cm)/ (25) = 4 cm = radius

Answer 3

Length of edge = a.
Diagonal of one face = a√2 = 4√3.
So a = 2√6
Length of each edge = 2√6 units.

Slant surface area = pi*R*L where R = radius = 6 units.
So, pi*R*L = 66*pi
6*L = 66
L = 11
So slant height = L = 11 units

Total area = Base area + Slant surface area
38*pi = pi*R^2 + pi*R*L
38 = R^2 + R*L
38 = 2^2 + 2*L
38 = 4 + 2*L
L = 17
Required slant height = 17 metre

Slant surface area = pi*R*L where slant height = L = 25 cm.
So, pi*R*L = 100*pi
R*L = 100
R*25 = 100
R = 4
Reqd. radius = 4 cm

Answer 4

What is the measure of one edge of a cube if the diagonal of one of its faces is 4√3 units?