Question

A reservoir has the shape of a pyramid with square base, with apex facing down. Its height is 5 m, a side from its base is 3 m and it is filled with water up to 4 m. (i) Let $x$ be the height in meters measured from the apex to the base of the reservoir. The weight in Newtons of a thin layer of water between $x$ and $x + Delta x$ is approximately $P(x)Delta x$. What is $P(x)$? $P(x) =$ FORMATTING: Express the answer using a formula. Recall that the density of water is $ ho = 1000 ext{ Kg/}m^3$ and the acceleration due to gravity on the earth's surface is $g = 9.8 ext{ m/}s^2$. (ii) The amount of work in Joules required to pump the thin water layer to the top of the reservoir is approximately $w(x)Delta x$. What is $w(x)$? $w(x) =$ FORMATTING: Express the answer as a formula. (iii) Using previous results, find, in Joules, the amount of work required to pump all of the water to the top of the reservoir. FORMATTING: If you round your answer, ensure that the round-off error is less than 1% of the value. Answer: Number Joules

          A reservoir has the shape of a pyramid with square base, with apex facing down. Its height is 5 m, a side from its base is 3 m and it is filled with water up to 4 m.
(i) Let $x$ be the height in meters measured from the apex to the base of the reservoir. The weight in Newtons of a thin layer of water between $x$ and $x + Delta x$ is approximately $P(x)Delta x$. What is $P(x)$?
$P(x) =$
FORMATTING: Express the answer using a formula.
Recall that the density of water is $
ho = 1000 	ext{ Kg/}m^3$ and the acceleration due to gravity on the earth's surface is $g = 9.8 	ext{ m/}s^2$.
(ii) The amount of work in Joules required to pump the thin water layer to the top of the reservoir is approximately $w(x)Delta x$. What is $w(x)$?
$w(x) =$
FORMATTING: Express the answer as a formula.
(iii) Using previous results, find, in Joules, the amount of work required to pump all of the water to the top of the reservoir.
FORMATTING: If you round your answer, ensure that the round-off error is less than 1% of the value.
Answer: Number Joules

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A reservoir has the shape of a pyramid with square base, with apex facing down. Its height is 5 m, a side from its base is 3 m and it is filled with water up to 4 m.
(i) Let x be the height in meters measured from the apex to the base of the reservoir. The weight in Newtons of a thin layer of water between x and x + Delta x is approximately P(x)Delta x. What is P(x)?
P(x) =
FORMATTING: Express the answer using a formula.
Recall that the density of water is ho = 1000 ext Kg/m^3 and the acceleration due to gravity on the earth's surface is g = 9.8 ext m/s^2.
(ii) The amount of work in Joules required to pump the thin water layer to the top of the reservoir is approximately w(x)Delta x. What is w(x)?
w(x) =
FORMATTING: Express the answer as a formula.
(iii) Using previous results, find, in Joules, the amount of work required to pump all of the water to the top of the reservoir.
FORMATTING: If you round your answer, ensure that the round-off error is less than 1% of the value.
Answer: Number Joules

Added by Kevin S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

07/03/2023

Step 1

The volume of a pyramid is given by V = (1/3)Bh, where B is the area of the base and h is the height. In this case, the base is a square with side length 3, so B = 3^2 = 9. The height is given as 5. Therefore, the volume of the reservoir is V = (1/3)(9)(5) = 15 Show more…

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A reservoir has the shape of a pyramid with square base, with apex facing down. Its height is 5 m, a side from its base is 3 m and it is filled with water up to 4 m. (i) Let x be the height in meters measured from the apex to the base of the reservoir. The weight in Newtons of a thin layer of water between x and x + Δx is approximately P (x) Δx . What is P (x) ? P (x) = FORMATTING: Express the answer using a formula. Recall that the density of water is ρ = 1000 Kg / m3 and the acceleration due to gravity on the earth’s surface is g = 9.8 m / s2 . (ii) The amount of work in Joules required to pump the thin water layer to the top of the reservoir is approximately w (x) Δx . What is w (x) ? w (x) = FORMATTING: Express the answer as a formula. (iii) Using previous results, find, in Joules, the amount of work required to pump all of the water to the top of the reservoir. FORMATTING: If you round your answer, ensure that the round-off error is less than 1% of the value. Answer: Number Joules

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Transcript

00:01 Hello everyone, in this problem we are given with a reservoir which is of the shape of a pyramid with a square base with apex facing down and its height is given as 5 meter and the side from the base is of 3 meter and the water is filled up to 4 meters of height.

00:18 So now the first part let us take x be the height in meters which is measured from the apex to the base of the reservoir so the weight in newtons of thin layer of a water between x and x plus del x is approximately p of x del x.

00:42 Now we need to find its p of x value so in order to find that p of x value so we can calculate the area of cross section at height x it would be a of x it would be 3 by 5 of x square multiplied by del x.

01:12 So here p of x would be equal to rho of 3 by 5 x square multiplied by del x.

01:23 Now let us move on to the next part of the equation where we need to calculate the amount of work in joules which required to pump the thin water layer to the top of the reservoir which is approximately wx of del x.

01:38 Now we need to calculate the wx which is the work done.

01:46 So for the next part we need to calculate the work done so the thin layer of water to the height x is given as w of x to be equal to p of x multiplied by g into x.

02:01 So here we have the p of x value as 3 by 5 x square multiplied by del x and g is the gravity which is of 9 .8 multiplied by 1000.

02:26 So in the next part of the equation so this is the required value of w of x so in the next part of question we need to calculate the work done so it would be w is from the given as from integral over 4 to 5 w of x d of x...

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