<span>Answer: To set up the integral, we divide the upper half of the aquarium into horizontal slices,
and for each slice, let x denote its distance from the top of the tank and ∆x denote
2
its thickness. (We choose horizontal slices because we want each drop of water in a
given slice to be the same distance from the top of the tank.) Using the formulae at
the beginning of this handout, we see that the work taken to pump such a slice out of
the tank is
work for a slice = W
= F · d
= (m · a) · d
= (ρ · V ) · a · d .
Since the length, width and thickness of the slice are given by 2 m, 1 m and ∆x m,
respectively, its volume is 2 · 1 · ∆x m3 = 2∆x m3
. Thus, the equation above becomes
work for a slice ≈
force
z }| {
mass
z }| {
(1000 kg/m
3
)
| {z }
density
(2∆x m
3
)
| {z }
volume
(9.8 m/s
2
)
| {z }
gravity
(x m)
| {z }
distance
= (1000)(9.8)(2)x · ∆x (kg · m/s
2
) · m
= (1000)(9.8)(2)x · ∆x N · m
= (1000)(9.8)(2)x · ∆x J .
Summing over our slices, this is
total work for top half of aquarium ≈
X(1000)(9.8)(2)x · ∆x J ,
where the sum is over the slices in the top half of the aquarium; that is, from distance
x = 0 to x = 1/2. As we refine our slices, this becomes the integral
total work = Z 1/2
0
(1000)(9.8)(2)x dx J
= (1000)(9.8)(2) Z 1/2
0
x dx J
= (1000)(9.8)(2)(1/8) J
= 2450 J .</span>
Problem #1:
We are approaching x = -1 from the left, and x is less than -1. Thus, use f1(x) = x - 5 instead of f2 or f3.
Just before x reaches -1, the value of f1(x) will be just a bit smaller than -1-5, that is, just a bit smaller than -6. Understanding that x can also equal -1, then the limit of f1(x) as x approaches -1 from the left is -6; it is defined.
Try the next problem. Share your work. I'd be happy to give you feedback on your efforts.
Answer:
25
Step-by-step explanation:
c^2=a^2+b^2
where c = hypotenuse
so, x^2 = 24^2+7^2
x^2 = 576 + 49
x^2 = 625
x = square root of 625
x = 25
60 is 40% of total trees.
total trees = 60 * 100/40 = 150
Answer is D.
The expression for the perimeter would be
2(x) + 2(2x+7)
which can be simplified to 2x+4x+14
which equals 6x+14
