Awater trough is 7 feet long, and its cross section is an equilateral triangle with sides 4 feet long. water is pumped into the trough at a rate of 4 cubic feet per second. how fast is the water level rising when the depth of the water is 1/2 foot?
( hint: first, what is the height h of an equilateral triangle of side length s? next, what is the area of an equilateral triangle in terms of the side length s? then write the area in terms of h. the volume of the water in the trough at time t is the product of the cross-sectional area with water and the length of the trough. )
a) what is the height h of an equilateral triangle of side length s?
b) the water level is rising at a rate of

a. An equilateral triangle main characteristic is that all the sides lenght are the same (s). To find the height (h), we could divide the triangle (as seen in the picture) and apply Pytagorean theorem.

Clearing the expression, we obtain:

b. Knowing the rate at which the volume is changing 4 , we can find the relation between the change in the volume and the height.

As we want to express the volume in terms of the height, we have to find the area in terms of height

Base=s=

Therefore,

Therefore the change in the volume with the height, will be the derivate of this expression

Knowing dV/dt=4 cubic feet per second, and h=1/2 foot, we can know dh/dt

Step-by-step explanation:

a) one student is chosen from classroom a, then a student is chosen from classroom b.  

step-by-step explanation:

independent events are ones in which the probability of one event does not affect the probability of the second event.

one student is chosen from classroom a; this does not affect the chances of choosing a student from classroom b.   this makes these independent events.

when one card is chosen from a standard deck and then set aside before a second card is chosen, the probability of the second card being chosen changes.   this makes them dependent events.

when one student is chosen from classroom a and then a second student is chosen from classroom a, the probability of choosing the second student changes.   this makes them dependent events.

when a letter is chosen from the alphabet and then a second letter is chosen, the probability of choosing the second letter changes.   this makes them dependent events.