The surface area of a soft drink with cylindrical shape is given by a formula SA=2×pi×r(r+h), where r cm is radius of can and h cm is the height. The height of a can of wood finish is 5 cm and its surface area is 250cm^2
Substitute values into formula to form quadratic equation using pronumeral, r.
Use quadratic formula to solve the equation and, hence, find radius of can. Round answer to 1 decimal place.
Calculate area of paper label around can. Round answer to nearest square centimeter.
SA=2π r(r+h) = 2π r^2 + 2πrh
2π r^2 + 2πr(5) = 250
πr^2 + 5πr -125 = 0
This is just a quadratic, where a = π, b = 5π, and c = -125
remember that π is just a constant, your good calculator has its value built in.
the paper area around the can would be the SA without the end circles.
Go for it, let me know what you get
Is it -8.92 or -22.5
To form the quadratic equation using the given formula, we substitute the values into the formula:
SA = 2πr(r + h)
Given: h = 5 cm and SA = 250 cm²
Substituting these values into the equation, we have:
250 = 2πr(r + 5)
Simplifying this equation, we now have a quadratic equation in terms of r.
250 = 2πr² + 10πr
Rearranging the equation and combining like terms, we get:
2πr² + 10πr - 250 = 0
Now we have a quadratic equation in the form ax² + bx + c = 0, where a = 2π, b = 10π, and c = -250.
To solve this quadratic equation, we can use the quadratic formula:
r = (-b ± √(b² - 4ac)) / 2a
Plugging in the values of a, b, and c:
r = (-(10π) ± √((10π)² - 4(2π)(-250))) / 2(2π)
Simplifying further:
r = (-10π ± √(100π² + 2000π)) / 4π
r = (-10π ± √(100π(π + 20))) / 4π
Now, we can calculate approximate values for the radius by substituting the π value and simplifying:
r = (-10π ± √(100π(π + 20))) / 4π
r = (-2.5 ± √(25(π + 20))) / π
Using the quadratic formula, we have two potential solutions for the radius. Since the radius cannot be negative, we ignore the negative solution and only consider the positive solution:
r = (-2.5 + √(25(π + 20))) / π
r ≈ 3.2 cm (rounded to 1 decimal place)
Therefore, the radius of the can is approximately 3.2 cm.
To calculate the area of the paper label around the can, we need to subtract the surface area of the can from the total surface area.
The surface area of the can is given by the formula SA = 2πr(r + h). We substitute the value of the radius we have found, which is approximately 3.2 cm, and the given height of 5 cm:
SA = 2π(3.2)(3.2 + 5)
SA ≈ 146.8 cm²
Since the total surface area is given as 250 cm², we subtract the surface area of the can:
Area of paper label = Total surface area - Surface area of the can
Area of paper label = 250 - 146.8
Area of paper label ≈ 103.2 cm²
Therefore, the area of the paper label around the can is approximately 103.2 cm² (rounded to the nearest square centimeter).