Question

Janine made a cylindrical vase in which the sum of the lateral area and area of one base

was about 3000 square centimeters. The vase had a height of 50 centimeters. Find the

radius of the vase. Explain the method you would use to find the radius.

Answer 1

Check the picture.

We have a cylinder of base radius R, and height H=50cm.

As we unfold it, we see that it is composed of a rectangle, and 2 circles.



The "lateral area" means the area of the rectangle, which has dimensions :

H by C, where C is the circumference of the circles.


Thus the lateral area =
`H \cdot C=H \cdot 2 \pi R=2 \pi HR \approx2 \cdot3.14 \cdot50R=314R` (centimeters)


The area of one base is the area of a circle with radius R, given by the formula:

`A_(base)= \pi R^2\approx3.14R^2` (square cm)


"the sum of the lateral area and area of one base was about 3000 square centimeters" means:


`314R+3.14R^2=3,000\\\\3.14R^2+314R-3,000=0\\\\3.14(R^2+100R-955.4)=0\\\\R^2+100R-955.4=0`

To solve the quadratic equation, we use the discriminant formula:

a=1, b=100, c=-955.4


`D=b^2-4ac=100^2-4(1)(-955.4)=10,000+3,821.6=13,821.6`


`√(D)= √(13,821.6)= 117.6`

the roots are :


`R_1= (-b+ √(D) )/(2a)= (-100+ 117.6 )/(2)= (17.6)/(2)=8.8`

and


`R_2= (-b- √(D) )/(2a)= (-100- 117.6 )/(2)\ \textless \ 0` which cannot be the radius, as it is a negative numbers.


Answer: 8.8 cm