Please help - Maths, maximum and minimum: 2000/x +x^2?
an open rectangular tank of height h meters with a square base of side x meters has a capacity of 500 cubic meters. Prove the survace area of the 4 walls and base will be 2000/x +x^2.
Find the value of x for this value to be a minimum
Answers:
First the proof
The capacity / volume of a tank would be area of base multiplied by height. In this case:
(x^2)h = 500
Therefore h = 500 / (x^2)
Surface area for 4 sides and base would be
Base x^2
One side xh, hence 4 sides 4xh
Total surface area x^2 + 4xh
Substitute h = 500 / (x^2) into x^2 + 4xh
This gives x^2 + 4x (500 / (x^2))
simplifies to give x^2 + 2000/x
SECOND PART - minimum part
it is a quadratic equation and I do not know what level of maths you are at but best way is through differentiation and using the second deriative to determine if it is a maximum or minimum.
http://algebrahelp.com/calculators/equat.
OK - well u know that x^2 times h is the volume which is 500, so..
x^2 times h = 500
u also kno that Surface Area (SA) is the base area plus 4 times x times h, so..
x^2 + 4xh = SA
They tell u that SA = 2000/x +x^2, so..
2000/x +x^2 = x^2 + 4xh
substitute h for a rearrangement of x^2 times h = 500, so that.
h = 500/x^2
therefore..
2000/x +x^2 = x^2 + 4x(500/x^2)
This can now be solved rearrange to get x, i wont do it for u cos u will get no enjoyment, but all u gotta do is think first of what the question tells u, hope this helps, as its got two divides either side u may end up with a negative answer, in most cases the negative can be removed and it will still work, but always double check.
Hope that helps, sorry i didn't complete it fully - but u would feel empty having it given to u, good luck.
First you know that volume for this tank is 500m^3 = hx^2; so you solve for h and you get h = 500/x^2.
Surface area:
bottom = x^2
all sides = 4hx
total surface area = x^2 + 4hx
substitute for h from above and you get:
total surface area = x^2 + 4(500/x^2)x = x^2 + 2000/x
Since "x" has to be positive (talking about the length and width of the tank) the smallest value of x I get is 10 m which would give you 300 meters squared for the surface area of the walls. (You can visually see it if you draw a graph with x on the bottom and surface area for the y-axis and plug in values from 1 to the square root of 2000).
ok, you have 5 walls in total. four side walls with surface x times h and one bottom wth surface x sqared.
That make the total surface area:
4hx + x^2
the total capacity is h times x times x (check your books). We also know that's 500. So:
hx^2 = 500. From here h = 500/x^2
You want to prove that the surface area is 2000/x + x^2, or:
4hx + x^2 = 2000/x + x^2
Well, if we substitute h with what we got wbove, we get:
4(500/x^2)x + x^2 = 2000/x + x^2
Do the maths and you'll see this is true.
To find the min, bear in mind that x^2 is always positive. so you want 2000/x to be the minimum. Negative values are smaller than positive and the only way to get a negative result from 2000/x is for x to be negative.
Obviously the higher the x by absolute value, the lower 2000/x will be. Like, 2000/2000 is 1, but 2000/1 is all the 2000. Because we are talking negatives, though, we'd like the highest possible negative. That is -1. With -1 as x you get -2000 + 1 = -1999. Can't beat that with any other value, try a few and see the logic.
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Find the value of x for this value to be a minimum
Answers:
First the proof
The capacity / volume of a tank would be area of base multiplied by height. In this case:
(x^2)h = 500
Therefore h = 500 / (x^2)
Surface area for 4 sides and base would be
Base x^2
One side xh, hence 4 sides 4xh
Total surface area x^2 + 4xh
Substitute h = 500 / (x^2) into x^2 + 4xh
This gives x^2 + 4x (500 / (x^2))
simplifies to give x^2 + 2000/x
SECOND PART - minimum part
it is a quadratic equation and I do not know what level of maths you are at but best way is through differentiation and using the second deriative to determine if it is a maximum or minimum.
http://algebrahelp.com/calculators/equat.
OK - well u know that x^2 times h is the volume which is 500, so..
x^2 times h = 500
u also kno that Surface Area (SA) is the base area plus 4 times x times h, so..
x^2 + 4xh = SA
They tell u that SA = 2000/x +x^2, so..
2000/x +x^2 = x^2 + 4xh
substitute h for a rearrangement of x^2 times h = 500, so that.
h = 500/x^2
therefore..
2000/x +x^2 = x^2 + 4x(500/x^2)
This can now be solved rearrange to get x, i wont do it for u cos u will get no enjoyment, but all u gotta do is think first of what the question tells u, hope this helps, as its got two divides either side u may end up with a negative answer, in most cases the negative can be removed and it will still work, but always double check.
Hope that helps, sorry i didn't complete it fully - but u would feel empty having it given to u, good luck.
First you know that volume for this tank is 500m^3 = hx^2; so you solve for h and you get h = 500/x^2.
Surface area:
bottom = x^2
all sides = 4hx
total surface area = x^2 + 4hx
substitute for h from above and you get:
total surface area = x^2 + 4(500/x^2)x = x^2 + 2000/x
Since "x" has to be positive (talking about the length and width of the tank) the smallest value of x I get is 10 m which would give you 300 meters squared for the surface area of the walls. (You can visually see it if you draw a graph with x on the bottom and surface area for the y-axis and plug in values from 1 to the square root of 2000).
ok, you have 5 walls in total. four side walls with surface x times h and one bottom wth surface x sqared.
That make the total surface area:
4hx + x^2
the total capacity is h times x times x (check your books). We also know that's 500. So:
hx^2 = 500. From here h = 500/x^2
You want to prove that the surface area is 2000/x + x^2, or:
4hx + x^2 = 2000/x + x^2
Well, if we substitute h with what we got wbove, we get:
4(500/x^2)x + x^2 = 2000/x + x^2
Do the maths and you'll see this is true.
To find the min, bear in mind that x^2 is always positive. so you want 2000/x to be the minimum. Negative values are smaller than positive and the only way to get a negative result from 2000/x is for x to be negative.
Obviously the higher the x by absolute value, the lower 2000/x will be. Like, 2000/2000 is 1, but 2000/1 is all the 2000. Because we are talking negatives, though, we'd like the highest possible negative. That is -1. With -1 as x you get -2000 + 1 = -1999. Can't beat that with any other value, try a few and see the logic.
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