Question

An inverted pyramid is being filled with water at a constant rate of 30 cubic centimeters per second . The pyramíd , at the top, has the shape of a square with sides of length 7cm and the height is 14 cm.

Answer 1

The answer is 30 cm/sec

EXPLANATION :

We have an inverted square pyramid with a square side of 7 cm and a height is 14 cm.

We need to find the area of the square at 2 cm from below.

Using similar triangles, we will express the side view as 2D :

We need to find the side of the square at 2 cm level.

The ratio of the sides of the smaller triangle and bigger triangle must be the same :

`\begin{gathered} \frac{\text{ smaller}}{\text{ bigger}}=(x)/(7)=(2)/(14) \\ \\ \text{ Solve for x :} \\ \text{ Cross multiply :} \\ 14x=7(2) \\ 14x=14 \\ x=(14)/(14)=1 \end{gathered}`

So the value of x is 1, then the side of the square at 2 cm level is 1 cm

The area of that square is :

A = 1 x 1 = 1 cm^2

The inverted pyramid is filled with water at a constant rate of Q = 30 cm^3 per second.

And we are asked to find the rate when the water level is 2 cm or when the area of the square is 1 cm^2 from the result we calculated above.

Recall the formula of rate :

`\begin{gathered} Q=AV \\ \text{ where :} \\ Q\text{ = constant rate in }(cm^3)/(sec) \\ \\ A\text{ = Area of the section in }cm^2 \\ \\ V\text{ = Velocity or rate in }(cm)/(sec) \end{gathered}`

We have the following :

`\begin{gathered} Q=30\text{ }(cm^3)/(sec) \\ \\ A=1\text{ }cm^2 \end{gathered}`

Using the formula above, the rate is :

`\begin{gathered} Q=AV \\ V=(Q)/(A) \\ \\ V=\frac{30\text{ }\frac{cm^{\cancel{3}}}{sec}}{1\text{ }\cancel{cm^2}} \\ \\ V=30\text{ }(cm)/(sec) \end{gathered}`