SOLUTION: in the question: an open box with a square Base is to be made out of given iron sheet of area 27 square metre. show that the maximum volume of the box is 13.5 cubic meters.? the s
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Question 1023300: in the question: an open box with a square Base is to be made out of given iron sheet of area 27 square metre. show that the maximum volume of the box is 13.5 cubic meters.? the surface is equals to the x^2 + 4xy.why? Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
volume = length * width * height.
since the length and the width is the same in a square base, then you can let side equal the length or the width and you get:
volume = side * side * height which results in:
volume = side squared * height.
if you allow x to represent the side and you let y represent the height, then you get:
volume = x^2 * y.
now surface area in a box with an open top is given by the following formula.
surface area = area of the base plus 4 times the area of the sides of the box.
the area of the base is equal to side squared which is equal to x^2.
the area of each side of the box is equal to side * height which is equal to x * y.
therefore the surface area of the box with an open top is equal to x^2 + 4 * xy which can also be shown as x^2 + 4xy.
your next question is why can't the volume of the box be greater than 13.5.
you know that the volume of the box = x^2 * y.
you also know that the surface area of the box = x^2 + 4xy.
you also know that the surface area of the box = 27 square centimeters.
you get 27 = x^2 + 4xy.
solve this equation for y and you get y = (27-x^2) / 4x.
replace y in the equation of volume = x^2 * y and you get:
volume = x^2 * (27 - x^2) / 4x.
simplify this equation and you get volume = x * (27 - x^2) / 4.
simplify further and you get volume = (27x - x^3) / 4.
solve for 4 times the volume and you get 4 times the volume = 27x - x^3.
if you use calculus, you can find the max/min point of this equation and you will see that it is when x = 3.
when x = 3, the equation becomes 4 * the volume = 27(3) - (3)^3 which becomes 4 * the volume = 54.
solve for volume and you get volume = 54 / 4 = 13.5.
this is the max/min point of the equation, which turns out to be the max point of the equation.
this is because this is the highest value of the equation when the value of x is greater than 0, as it has to be.
you can also solve the equation graphically as shown below:
the equation you are graphing is y = (27x - x^3) / 4
in this graph, y represents the volume and x represents the the values of the area of the base times the height in terms of x only as derived above.
fyi, the calculaus solution says that the first derivative of the equation of 27x - x^3 is 27 - 3x^2.
set this equal to 0 and you get 27 - 3x^2 = 0.
add 3x^2 to both sides to get 27 = 3x^2.
divide both sides by 3 to get 9 = x^2
solve for x to get x = plus or minus 3.
x has to be positive, so x = 3.
