Bob's Barricades rents orange plastic cones to use as barricades in road construction. Each cone is 24 inches high with a diameter of 14 inches. A gallon of orange paint will cover approximately 400 square feet. About how many whole gallons of paint will Barb need to paint the lateral sides only of the 300 cones.
I will assume that we paint only the lateral surface of each cone
SA = πrs, where s is the slant height
http://math.about.com/od/formulas/ss/surfaceareavol_2.htm
so s^2 = 7^2 + 24^2
s = √625 = 25
So each cone has a lateral SA of π(7)(25) or 175π inches^2
300 cones have 52500π inch^2
number of gallons of paint
= 52500π/400
=412.33
but we can't buy a partial gallon, so
413 gallons are needed
check my arithmetic
It is 3 gallons
To find out how many whole gallons of paint Barb will need to paint the lateral sides of the 300 cones, we need to calculate the total area of the lateral sides of all the cones and then divide it by the coverage of one gallon of paint.
First, let's calculate the lateral surface area of one cone. The lateral surface area of a cone can be calculated using the formula:
Lateral Surface Area = πr × l
Where:
r is the radius of the base of the cone (half the diameter)
l is the slant height of the cone
Given that the diameter of each cone is 14 inches, the radius (r) would be 14/2 = 7 inches.
To find the slant height (l), we can use the Pythagorean theorem. The slant height can be calculated as:
l = √(r^2 + h^2)
Where:
h is the height of the cone
Given the height of each cone is 24 inches, the slant height (l) would be:
l = √(7^2 + 24^2) = √(49 + 576) = √625 = 25 inches
Now we can calculate the lateral surface area of one cone as:
Lateral Surface Area = π × 7 × 25 = 175π square inches
Since we need to paint the lateral sides of 300 cones, we multiply the lateral surface area of one cone by 300 to get the total lateral surface area:
Total Lateral Surface Area = 175π × 300 = 52500π square inches
To convert square inches to square feet, we divide the total lateral surface area by 144 (since there are 144 square inches in a square foot):
Total Lateral Surface Area in square feet = (52500π) / 144
Now we divide the total lateral surface area in square feet by the coverage of one gallon of paint (400 square feet) to find out how many gallons of paint Barb will need:
Number of gallons of paint = (52500π) / (144 * 400)
Using a calculator, we can multiply 52500 by π, then divide the result by (144 * 400) to obtain the answer in gallons.