I forget how to do these type of word problems.
Assume you working on the house pictured below. The length of the house is 50 feet, the width is 30 feet, and from the ground to the gable (the height of the rectangle) is 10 feet. The height of the triangular prism on top is 6.5 feet.
The owner of the house wants to estimate the amount of paint needed for the house. If one gallon of paint is expected to cover 400 square feet of surface, how many full gallons of paint will be needed?
area of walls: 2(50*10 + 30*10) = 1600
area under roof: 2(1/2 * 30 * 6.5) = 195
total area: 1795 ft^2
gallons = 1795ft^2 / 400ft^2/gal = 4+ gals, or 5 gallons
Thank you Steve this helped me!
To find the amount of paint needed to cover the surface area of the house, you will need to calculate the surface area of each component (rectangle and triangle) and then add them together.
Let's start by finding the surface area of the rectangle. The formula for calculating the surface area of a rectangle is length multiplied by width.
In this case, the length of the rectangle is 50 feet and the width is 30 feet. So, the surface area of the rectangle can be calculated as follows:
Surface Area of Rectangle = Length * Width
= 50 feet * 30 feet
= 1500 square feet
Next, let's calculate the surface area of the triangular prism on top. The formula for finding the surface area of a triangular prism is as follows:
Surface Area of Triangular Prism = (Base * Height) + (Perimeter of Base * Height) / 2
In this case, the base of the triangle is the same as the width of the rectangle, which is 30 feet. The height of the triangle is given as 6.5 feet.
To find the perimeter of the triangle, you need to add all three sides together. However, since the triangle is isosceles, only two sides need to be added together and then multiplied by 2. By using the Pythagorean theorem, you can find the length of the third side.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Hence, using the values given:
Third Side^2 = (Base / 2)^2 + Height^2
= (30 feet / 2)^2 + 6.5 feet^2
= 225 square feet + 42.25 square feet
= 267.25 square feet
Taking the square root of 267.25 square feet, we get:
Third Side ≈ 16.34 feet (approximated to two decimal places)
So, the perimeter of the triangle is: Perimeter = (Base * 2) + Third Side ≈ (30 feet * 2) + 16.34 feet ≈ 76.34 feet
Now, we can substitute these values into the formula for finding the surface area of the triangular prism:
Surface Area of Triangular Prism = (Base * Height) + (Perimeter of Base * Height) / 2
= (30 feet * 6.5 feet) + (76.34 feet * 6.5 feet) / 2
= 195 square feet + 2489.1 square feet / 2
≈ 195 square feet + 1244.55 square feet
≈ 1439.55 square feet
Finally, to calculate the total surface area, we need to add the surface areas of the rectangle and the triangular prism:
Total Surface Area = Surface Area of Rectangle + Surface Area of Triangular Prism
= 1500 square feet + 1439.55 square feet
≈ 2939.55 square feet
Now, we can determine the number of gallons needed. We know that one gallon of paint will cover 400 square feet of surface area. So, we divide the total surface area by the coverage per gallon:
Number of Gallons Needed = Total Surface Area / Coverage per Gallon
= 2939.55 square feet / 400 square feet
≈ 7.35 gallons (approximated to two decimal places)
Therefore, approximately 7.35 full gallons of paint will be needed to cover the entire surface area of the house.