Math help please?

Question

Tell how u got it.
1. what is the radius, in inches, os a right circular cylinder if its lateral surface are is 3.5 square inches and its volume is 3.5 cubic inches?

2. a circle is circumscribed about a egualateral triangle with side lenghts od 9 units each. what is the area of the circle, in square units? express in pi form

Answer 1

1. volume of a cylinder is (Pi)(r^2)h.
lateral surface area is 2(Pi)(r)h.
since we are given that both =3.5 we can set them equal to each other. then, if we divide both sides by (pi)(r)h we end up with r=2

2. this one would b easier to explain with a diagram. I'll try my best without one. if u take the center of the circle, and draw a line to each vertex of the triangle, you'll end up with 3 isosceles triangles. the base of each triangle will be 9 units and the other two sides will be the same value as the radius of the circle which we will call r. now, take those lines and extend them so that they form the perpendicular bisectors of the triangle. you end up with 6 30-60-90 triangles. the sides of a 30-60-90 triangle are a, a/2 and a(sqrt(3))/2. since we knew that the side of the original triangle was 9. we can see that a(sqrt(3))/2=9/2. then a(sqrt(3))=9 and a=3sqrt(3). and if you look at the diagram u hopefully drew, a is the radius of the circle! so pi*r^2= 27pi

Answer 2

2. Draw the figure with one vertex of the triangle at the top of the circle then draw a radius that hits the bottom left vertex. Also draw an altitude of the triangle from the top vertex to the midpoint of the bottom. At the bottom left is a little 30-60-90 right triangle with long leg 4.5. The short leg of a 30-60-90 is the long leg divided by √ 3 and the hypotenuse (which is the radius which you need for the circle's area) is twice the short leg so the radius is 2(4.5)/ √ 3 or 9 / √ 3.

Now do pi r squared = pi (81/3)

1. LSA = 2 pi r h and V = pi r^2 h. Since they equal the same number, 2 pi r h = pi r^2 h

divide both sides by pi r h and you'd have

2 = r

Answer 3

The lateral surface area is 2 x pi x radius x height
The volume is pi x (radius)^2 x height.
Since they are numerically equal, you can set the above relations equal to each other.

Draw two angle bisectors from two angles to their intersection, which is at the center of the circle. You will form a 30/30/120 triangle. Now draw an angle bisector from the 120 deg angle to the base of the triangle. Now you have a 30/60/90 triangle, whose longer non-hypotenuse side is 4.5 units. For such a triangle, the ratio of the hypotenuse to the longer side is 2 to sqrt(3). So....
2/sqrt(3) = Radius/4.5, or the radius = 9/sqrt(3) or
3 sqrt(3). The rest is yours.

Answer 4

Remember ;

Lateral surface = Perimeter of the base x height of the cylinder. and

Volume = Area of the base x height of the cyl.

Now if R is the radius, h is the height of the cyl. --------

Lateral surface is ( 2.Pi.R.h ) = 3.5
ie Pi. R.h = 1.75 ................................ (1)

Again Volume of the cylinder
= Pi.R^2.h . which can be re-written as
= ( R . Pi.R.h ) ----- Splitting R^2 as R.R
= ( R . 1.75 ) ------------ Using relation (1)

So that R . 1.75 = 3.5 (given)
Hence R = 2 in. .... Ans