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 = [tex]H \cdot C=H \cdot 2 \pi R=2 \pi HR \approx2 \cdot3.14 \cdot50R=314R[/tex] (centimeters)
The area of one base is the area of a circle with radius R, given by the formula:
[tex]A_{base}= \pi R^2\approx3.14R^2[/tex] (square cm)
"the sum of the lateral area and area of one base was about 3000 square centimeters" means:
[tex]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[/tex]
To solve the quadratic equation, we use the discriminant formula:
a=1, b=100, c=-955.4
[tex]D=b^2-4ac=100^2-4(1)(-955.4)=10,000+3,821.6=13,821.6[/tex]
[tex] \sqrt{D}= \sqrt{13,821.6}= 117.6[/tex]
the roots are :
[tex]R_1= \frac{-b+ \sqrt{D} }{2a}= \frac{-100+ 117.6 }{2}= \frac{17.6}{2}=8.8[/tex]
and
[tex]R_2= \frac{-b- \sqrt{D} }{2a}= \frac{-100- 117.6 }{2}\ \textless \ 0[/tex] which cannot be the radius, as it is a negative numbers.
Answer: 8.8 cm
