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

, and

for

step-by-step explanation:

we can analyze the pattern at which the terms increase. let us denote , , , so that we can call

note the following:

the pattern: to get the next term value, we just add the next term number (e.g. 1st, 2nd, to the previous term value (term values are the given numbers in the question, 1, 3, 6, 10, 15,

from , to get to , we add

from , to get to , we add

etc.

we need to form a recursive formula, a formula that has the next term as a function of the previous term. so a possible recursive formula can be

, and for