Problem 44 - Algebra 2

Question

Water Supply The velocity of the water that flows from an opening at the base of a tank depends on the height of water above the opening. The function \(v(x)=\sqrt{2 g x}\) models the velocity \(v\) in feet per second where \(g\) , the acceleration due to gravity, is about 32 \(\mathrm{ft} / \mathrm{s}^{2}\) and \(x\) is the height in feet of the water. Find the inverse function and use it to find the depth of water when the flow is 40 \(\mathrm{ft} / \mathrm{s}\) , and when the flow is 20 \(\mathrm{ft} / \mathrm{s}\) .

Step-by-Step Solution

Answer

The depth of the water is 25 ft when the flow is 40 ft/s and it is 6.25 ft when the flow is 20 ft/s.

Step 1: Construct the inverse function

The initial function is \(v(x)=\sqrt{2gx}\). To find the inverse function, interchanging the roles of x and v yields \(x(v)=\sqrt{2gv}\). Squaring both sides allows us to solve for x: \(x^{2}(v)=2gv\). Solving for x gives the inverse function: \(x(v)=\frac{v^{2}}{2g}\) .

Step 2: Calculate the depth of water when the flow is 40 ft/s

To find out the depth of the water when the flow is 40 ft/s, substitute v=40 ft/s and g=32 ft/s² into the inverse function: \(x(40)=\frac{(40)^{2}}{2*32} = 25\) ft.

Step 3: Calculate the depth of water when the flow is 20 ft/s

Similarly, to find the depth when the flow is 20 ft/s, substitute v=20 ft/s into the inverse function: \(x(20)=\frac{(20)^{2}}{2*32} = 6.25\) ft.

Key Concepts

Water Flow

Imagine water pouring from a tank through a small hole at the bottom. The speed or velocity at which this water escapes isn't random; it depends on how high the water is above the opening. This principle is essential in understanding fluid dynamics and is particularly useful in many engineering and physics applications. The water velocity from the tank can be modeled with the formula \( v(x) = \sqrt{2gx} \), where \( v \) is the velocity, \( g \) is the acceleration due to gravity, and \( x \) is the height of the water.

With water flow, as the height \( x \) increases, velocity rises because there is more gravitational energy. Conversely, when \( x \) is smaller, the velocity decreases. This relationship helps us control water systems in practical ways, like designing the ideal height for tanks to maximize pressure.

Velocity Equation

The velocity equation \( v(x) = \sqrt{2gx} \) tells us the speed that water takes when exiting a tank, influenced heavily by gravity. To unpack this, remember that velocity refers to how quickly something moves. Water gains velocity as it descends because the force of gravity pulls it faster down the slope of the tank.

This equation matters because, in real-world scenarios, managing water systems involves controlling flow rates. Engineers must often calculate how fast water flows to ensure pipes do not burst or overflow due to excessive pressure. By understanding the velocity equation, they can design systems that either slow down or speed up water flow as necessary.

• The variable \( v \) stands for the velocity of water.
• The constant \( g \) represents gravity, approximated at 32 ft/s² on Earth.
• The variable \( x \) indicates the water height above the exit point.

Gravity in Physics

Gravity is the invisible force that pulls objects towards each other. On Earth, it defines how things fall and affects every moving object, including flowing water. Its role in fluid mechanics is pivotal, determining not only how fast water flows from a tank but also impacting countless natural and industrial processes.

When we talk about the acceleration due to gravity (\( g \)), we're referring to how quickly an object speeds up as it falls. Here, it's crucial for determining the velocity of water as it exits the tank. As gravity pulls water downward, it accelerates the flow rate, increasing velocity, which can be calculated using the equation \( v(x) = \sqrt{2gx} \).

In broader physics, gravity impacts not just water flow but orbits of planets, tides in the ocean, and our own day-to-day experiences of moving in the world. By grasping gravity's role in motions, students can appreciate its fundamental importance and applications.