A rectangular sheet of metal, of area 440 sq.cm is folded into a cylinder, to enclose a volume of 1540 cu.cm. A circle whose diameter is equal to the width of the rectangular sheet is cut from the sheet. Find the ratio of the area of the uncut region to the area of the cut region?
let the radius of the cylinder formed be x cm
then the length of the rectangle is 2pi(x)
let the width of the rectangle by y cm
pi(x^2)(y) = 1540
2pi(x)(y) = 440
divide first by second
x/2 = 3/5
x = 7
so area cut out = pi(7)^2 = 49pi
uncut area = 440 - 49pi
ratio of uncut:cut
= (440-49pi)/(49pi)
To solve this problem, we need to break it down into steps:
Step 1: Find the dimensions of the rectangular sheet.
Let's assume the length of the rectangular sheet is L cm and the width is W cm.
Step 2: Use the given information to set up equations.
We are given that the area of the rectangular sheet is 440 sq.cm:
L * W = 440 -- Equation 1
We are also given that the volume of the cylinder formed after folding the sheet is 1540 cu.cm. The volume of a cylinder is given by the formula:
Volume = π * r^2 * h
Since the cylinder is formed by folding the rectangle, the height of the cylinder is equal to the width of the rectangular sheet (W cm).
We can find the radius (r) of the cylinder by using the formula for the circumference of a circle:
Circumference = 2 * π * r
Since the diameter of the circle is equal to the width of the rectangular sheet (W cm), the circumference of the circle is W cm.
So we have:
2 * π * r = W -- Equation 2
We can also relate the area of the circle to the area of the rectangular sheet:
Area of circle = π * r^2
Area of rectangle = L * W
Area of uncut region = Area of rectangle - Area of circle
Step 3: Solve the equations to find the dimensions and areas.
From Equation 2, we can solve for r:
r = W / (2 * π)
Substituting the value of r in terms of W into Equation 1, we get:
L * W = 440
L = 440 / W
Substituting the values of L and r into the equation for the area of the uncut region, we have:
Area of uncut region = (L * W) - (π * r^2)
Area of uncut region = (440 / W) * W - π * (W / (2 * π))^2
Area of uncut region = 440 - (W^2) / 4
Step 4: Find the area of the cut region.
The area of the cut region is equal to the area of the circle:
Area of cut region = π * r^2
Area of cut region = π * (W / (2 * π))^2
Area of cut region = (W^2) / 4
Step 5: Find the ratio of the area of the uncut region to the area of the cut region.
Ratio = (Area of uncut region) / (Area of cut region)
Ratio = (440 - (W^2) / 4) / ((W^2) / 4)
Now you can simplify the expression and evaluate the ratio.