A trapezoidal plate with bases 4 m and 8 m is partially submerged vertically in water so that the top is 2 m above the surface and the bottom is 2 m below the surface. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Round your answer to the nearest whole number. Use 9.8 m/s2 for the acceleration due to gravity. Recall that the weight density of water is 1000 kg/m3.)

F = 91466.667 N

Explanation:

Solution:-

- First you need to find an area equation for the triangle section of the trapezoid where the width is increasing from 4m to 8m, keeping in mind that we're only interested in the area that is submerged in water (2m).

- Set up the equation so that you divide the total height (4m) by the maximum width of this section (2m since (8m - 4m)/2, there's 2m on each side). This is equal to the line from the base to the surface of the water (2 - x) divided by a, which is the width at 2-x.

                                ( 2 - x ) / a = 4 / 2

                                a = ( 2 - x ) / 2

- Now we can find the area(A) of the trapezoid, which is the base added to double the width of the triangle (a), since the triangle is on both sides of the trapezoid. To find the area, we will multiply all of that by the change in height, which is Δx.

                               A = ( 4 + 2*a) * Δx

                               A = ( 4 + 2*((2 - x ) / 2)) * Δx

The force is calculated by taking the definite integral on the interval submerged in water of the pressure multiplied by gravity, the change in depth (2-x) and the area equation.

                              F=∫ p*g*(2−x). dA

Where, p : Density of water = 1000 kg/m^3

            g : gravitational acceleration = 9.8 m/s^2

- We calculate:

                             F = (9.8)*(1000) ∫ ( 2 - x ) * ( 10 - x ) / 2

- Set up limits from 0 - > 2. And Evaluate:

                             F = (9800)*(28/3)

                             F = 91466.667 N

carbon is found in all living things. carbon atoms move constantly through

living organisms, the oceans, the atmosphere, and the earth’s crust in what is

known as the carbon cycle. the directions taken by carbon atoms through this

cycle are very complicated and can take millions of years to make a full circle.

all animals, from humans to the dinosaurs are part of the carbon cycle. when

animals eat food, they get carbon in the form of carbohydrates and proteins.

in animals, oxygen combines with food in the cells to produce energy for daily

activity and then gives off carbon. the carbon combines with oxygen to form

carbon dioxide (co2) and is released back into the atmosphere as a waste

product when animals breathe and exhale.

from 145-65 million years ago, earth was much

hotter than today and covered with dense,

tropical swamp forests. the trees and other

plants were immense and

provided an endless supply of

food for the giant animals that

roamed the land. somewhere in

the air above one of these

forests, a lone carbon atom has joined

up with two oxygen atoms to form a molecule of co2.

the co2 molecule was sucked into the tiny holes (stomata) on the leaf of a

fern plant and joined with sunlight, chlorophyll and water to make food and

energy in the plant’s cells through photosynthesis. the oxygen (o2) from the

co2 molecule was sent back into the atmosphere; the carbon atom (c) was

detached and used to make a molecule of sugar.

if that carbon atom had been eaten instead, a totally

different situation would have developed. let’s say

that a giant plant-eating dinosaur (herbivore) ate the

fern for breakfast, and swallowed the carbon in the

fern in the form of carbohydrates and proteins. in

the process of respiration, oxygen combined with

the carbohydrates and proteins in the dinosaur’s cells

to provide energy for its daily activity.  

c because the action was the roller coaster reaches bottom of hill (action) reaction would be blood tends to drain away from head

hope this !