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Answer :
a) The amount of sheet metal required to build the trough it means the surface area is 615 +[tex]120\sqrt{34}[/tex].
b) The volume of the trough in cubic inches is 900[tex]in^{3}[/tex]
c) A cubic inch is about 0.004 33 gallons. 14.75 litres of water it can hold
What is surface area ?
The sum of all an object's faces makes up the surface area of a three-dimensional object. When wrapping, painting, and eventually building things to achieve the finest design, we apply the idea of surface areas of various items in real life.
Given that : The triangular face is a right triangle with a base of 3 inches and height of 5 inches. The trough runs the length of the barn, which is 120 feet long.
a) The amount of sheet metal required to build the trough it means the surface area which is equals to ( two rectangular areas + two triangular areas)
So, S = 2×[tex]\frac{1}{2}[/tex]×3×5 + 5×120 +[tex]\sqrt{3^{2}+\5{2} }[/tex] ×120
= 15 + 600 +[tex]120\sqrt{34}[/tex]
=615 + [tex]120\sqrt{34}[/tex] [tex]in^{2}[/tex]
b) V = SH
=[tex]\frac{1}{2}[/tex] ×3×5×120
=900[tex]in^{3}[/tex]
c)A cubic inch is about 0.004 33 gallons
so, 900 ×0.00433 =3.897 gallons
3.897 gallons = 3.897 ×3.785412
≈14.75L
14.75 litres of water it can hold.
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Final answer:
The total sheet metal required for the trough is 7215 square inches. The volume of the trough is 10800 cubic inches, which can hold approximately 46.764 gallons or 177 liters of water.
Explanation:
Solution for the Trough Sheet Metal and Volume Calculations
To calculate the amount of sheet metal required to build the triangular section water trough for a poultry farm, we must consider the surface area of the trough minus the base, as it's typically placed on the ground and doesn't require sheet metal. Since the trough has a right-angled triangular cross-section with a base of 3 inches and height of 5 inches, its area can be calculated using the formula for the area of a triangle, which is 1/2 * base * height. The length of the trough is given in feet, so we must convert it to inches (1 foot = 12 inches) before we can find the total surface area.
The area of the triangular face is 1/2 * 3 in * 5 in = 7.5 in^2. There are two triangular faces and a rectangular strip that runs the length of the trough, which would be 120 ft * 12 in/ft = 1440 inches long. The rectangle's width is the same as the height of the triangle, 5 inches. Therefore, we have:
1 rectangular face: 1440 in * 5 in = 7200 in^2
To find the total sheet metal required, we add these two areas together to get 7215 in^2.
Next, to determine the volume of the water trough, we use the formula for the volume of a rectangular prism because the trough extends along its length as a prism with a triangular cross-section. The volume is cross-sectional area * length, so:
Volume of the trough = Area of the triangle * Length of the trough = 7.5 in^2 * 1440 in = 10800 in^3.
Finally, to convert the volume to gallons and then to liters, we multiply 10800 in^3 by 0.00433 gallons/in^3 to get gallons and then multiply by 3.78541 to convert gallons to liters:
Total gallons = 10800 in^3 * 0.00433 gal/in^3 = approximately 46.764 gallons.
Total liters = 46.764 gallons * 3.78541 L/gallon = approximately 177 L.
