Bike race Theo and Sasha start at the same place on a straight...

Question

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1

a. Graph the velocity function for both riders.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Two trains, train X and train Y, depart from the same station. Their velocities in meters per second are train X, the subscript X of T is equal to 5, or T is greater than or equal to 0, and train the subscript Y of T is equal to 6T or 0 is less than or equal to T and T is less than or equal to 3, and V subscript YF T is equal to 18. For T is greater than 3, where T is in seconds, graph the velocity functions for both trains. Awesome. So it appears for this particular prom we're asked to take all the information that is provided to us by the prom itself, and we're asked to graph the velocity functions for both trains. So we'll have a curve for train X, and then we'll have a curve for train Y. For the velocity functions for both of these trains. So now that we know we're trying to solve 4. Let's take a moment to investigate this blank graph that is provided to us by the problem itself. We have our Y axis which denotes the velocity VFT, which starts from the origin 00 and goes up to 20. Then we have our X axis which extends from -1 to 5. Fantastic. And now that we have an idea of what our blank graph looks like, because we're going to be using this blank graph to plot our two curves. Our first step that we're going to need to take in order to solve this particular problem is we need to identify the velocity functions for both trains. So for train X, as we're told, VX of T will be equal to 5. And this is for T is greater than or equal to 0, and for train Y, however, we have V Y of T is equal to 6 T. 40 is less than or equal to T and T is less than or equal to 3, and Vy of T is equal to 18, or T is greater than 3. Fantastic. We also need to recognize that VX of T is a constant function, so I'll call it CF for short for a constant function. So because VX of T is a constant function, its graph will be a horizontal line at V equals 5. For all, T is greater than or equal to 0. Fantastic. So with that in mind, let us now focus our attention on train Y. So for train Y, we need to note that V Y T is a linear function, which I'll call LF for short. So VY of T is a linear function. So when we focus on Vy of T is equal to 60, this will be from T equals 0 to T equals 3. Fantastic. So with that in mind, Our next step now is we need to find VY of 0 and VY of 3. So that means we're going to be plugging in a value of 0 in for T and we're going to be plugging in a value of 3 in for T into our expression. So VY of 0 is equal to 6 multiplied by 0, which is equal to 0 m per second. And VY of 3 is equal to 6 multiplied by 3, which is equal to 18 m per second. Fantastic. So therefore, as a result, its graph will be a straight line starting at the origin point 0.0, and it increases, so I'll put an arrow pointing up and increases to V equals 18 at T equals 3. So then we'll reach the coordinate. 3.18. Fantastic. So with that in mind now, we need to focus our attention on for the condition. T is greater than 3, that will mean that Vy of T becomes a constant function, fantastic. So because it becomes a constant function, that means that Vy of T is equal to 18, so it's graph will be a horizontal line at V equals 18 for the condition where, so this is for the condition. Where T is greater than 3, fantastic. So now, at this point, we can plot both velocity functions, and when we do, we will determine that when we draw, so what we're gonna be doing when we go to plot our Two curves is that we will draw a horizontal line for train X at V equals 5, and for train Y we will draw the increasing line from 0.0 to 3.18, and then we will draw horizontal line at V equals 18 for T is greater than 3. And when we do that, our graph should look something like the following. So I've gone ahead and created a digital graph for us to refer to. And it will look as follows. So we have our Y-axis, which is VFT, and we have our first curve. In red for V X of T, so VX of T, so that's for train X is a is a horizontal line at V equals 5. Then we have our second curve represented in blue for train Y for VY of T. And as you can see, it starts at the origin point 0.0, and it goes up till it reaches the coordinate point 3.18, and then it goes straight out. To the right to form that horizontal line for where T is greater than 3 for V equals 18. And that's it. That is our final answer. All right, we did it. So we have officially graphed both of our curves for the velocity functions for both trains, and that's our final answer we were ultimately trying to solve for. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.
Theo: vT(t)=10, for t≥0
Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1

a. Graph the velocity function for both riders.

Key Concepts

Velocity as a Function of Time

Piecewise Functions

Graphing Functions

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