A can of beans has a surface are of 367^2. It's height is 18cm. What is the radius of the circular top?

Question

 I have been having trouble with this question for awhile thanks for the help

Answer 1

Very simple!

 

What is a can of beans? A can is a Cylinder shape, right?

 

Surface Area of a Cylinder = 2(Pi)r^2 + 2(Pi)rh  (that is, 2 Pi r-squared + 2 Pi r h)

 

How does that come about? Top of the cylinder is a 'disc' or a circle of area = (Pi)r^2 (that is, Pi r-squared)

                                         Bottom of the cylinder is again a disc or circle of area = (Pi)r^2

                                         Side area of the cylinder is (circumference)Xheight, that is, 2(Pi)rh

 

                                         Add all 3 above for surface area of the can: S = 2(Pi)r^2 + 2(Pi)rh

 

 

The problem says,  2(Pi)r^2 + 2(Pi)rh = 367 cm2 (that is, 367 centi-meter-squared)

 

                             Now, h = 18 cm (says in problem)

 

                             So,  2(Pi)r^2 + 2(Pi)r18 = 367  (I have substituted 18 cm for h)

 

                             We have to solve for 'r' (radius). We know that Pi = 3.14

 

                             Simplifying (that is substitute Pi = 3.14, in above expression), we get,

 

                                        r^2 + 18r - 58.44 = 0  (this is a quadratic equation in 'r')

 

                              This is a quadratic equation of the form ax^2 + bx + c, where,

                                                                               a = 1, b = 18, and c = -58.44

 

                               Solution for this equation is:  [-b (+/-) sq.rt. (b^2-4ac0]/2a

 

                               In our case,  r = [-18(+/-) sq.rt. (18^2 - 4(1)(-58.44)]/2(1)

 

                                             or r = [-18 +/- 23.6]/2 = 5.6/2 = 2.8 

 

                                                r = 2.8 cm  is the answer

 

When you do long calculations like this, it is prudent to check if your answer is correct. Very easy to do - - that is to be certain that our answer r = 2.8 cm.

 

All you do is substitute the value of r (= 2.8 cm) and h (=18) in the expression for surface area on top:

 

         It will simplyfy to:  S (surface area) = 49.2 + 316.5 = 365.7 (rounded to 366 cm^2), which is close enough to 367 cm^2 (see problem). The very slight error may come from the value of Pi. I used Pi = 3.14 rather than Pi = 3.1429.

 

 

                                

 

 

 

                               

 

 

 

Answer 2

A can takes on the shape of the cylinder.  The surface are of a cylinder is

 

SA = 2πr² + 2πrh

 

where:

r = radius

h = height

 

Plug in the values into the formula.

 

SA = 2π(r² + rh)

SA = 2π(r² + rh)

367 / 2π = r² + 18r

58.41 = r² + 18r

 

Subtract 58.41 on both sides of equation.

0 = r² + 18r - 58.41

 

Use the quadratic formula to solve for r.

r = (-b ± √(b² - 4ac))) / 2a

 

where:

a = 1

b = 18

c = -58.41

Plug in these values into the formula.  You will get two values of r because of the plus/minus sign.  Accept the positive value of r.

Answer 3

A can of beans is cylinder

 

Surfacr area of cylender

 

A=2πrh+2πr²

 

given

A=367 square cm

 

h= 18 cm

 

367=2πr(18)+2πr²

 

367=2πr(18+r)

 

367/2πr=18+r

 

π=3.14

 

58.41/r=18 +r

 

by cross multiplication

 

58.41=r(18+r)

 

58.41=18r+r²

 

r²+18r-58.41=0

 

using quadratic formula

r=2.80 ,r=-20.80

 

as radius cannot be negative

 

r=2.80 cm