An aquarium 4 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3). Show how to approximate the required work by a Riemann sum. Express the work as an integral.
Evaluate the integral.

Workdone = 4900joules

Step-by-step explanation:

Work-done = force x distance

Force = mass x acceleration

If we suppose that the aquarium is at height 0 at the top and height 1 at the bottom (since it's 1m deep), then we need to lift the slide at height x by x metres.

Mass = Density x Volume, therefore force needed to lift the slice is given is

F = density*volume*∆x*acceleration

Since we the mass changes as we take out water then there will be a change in height, for this reason we have ∆x.

Acceleration is due to gravity, therefore g=9.8m/s²

Density = 1000kg/m³

Volume = 4*1*1 = 4m³

F = 1000*4*∆x*9.8 = 39200∆x

Since our distance from the top of the aquarium to the bottom is x metres, then

Work-done = 39200x∆x

To find the total workdone, we find the integral of 39200xdx and the limit will be from 0 to 0.5 since we are to pump halfway of the aquarium. Integrating that we have

19600*x² limit from 0 to 0.5

19600(0.5² – 0²) = 19600(0.25 – 0) = 19600(0.25) = 4900joules

the anwser is d) 75°

step-by-step explanation:

region b will make the inequalities y < 2x + 1 and y > -x - 2 true ⇒ answer a

step-by-step explanation:

* lets study the graph and the inequality to solve the question

- the inequality y > -x - 2

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = -2 that means the line intersect the y-axis at point (0 , -2)

∵ the sign of the inequality is >

∴ we shad the part over the line

  region a and b

- the inequality y < 2x + 1

∵ the form of the inequality is y > mx + c , m is the slope of the line

  and c is the y-intercept (the line intersect y-axis at point (0 , c)

∴ c = 1 that means the line intersect the y-axis at point (0 , 1)

∵ the sign of the inequality is <

∴ we shad the part under the line

  region b and c

∵ the common region between the two lines is region b

∴ region b will make the inequalities y < 2x + 1 and y > -x - 2 true