domain / range Qs for rate r(t) = 800 - 40t, 0 < t < 30

[santaclaus]

1. Let t be in seconds and let r(t) be the rate< in gallons/second, at which water enters a reservoir. We are given that, for t 0<t<30,: r(t) = 800 - 40t

1. evaluate the expressions r(0), r(15), r(25) and explain their physical significance.

2. if you were to graph y=r(t) for 0<t<30, labeling the intercepts. What is the physical significance of the slope and the intercepts?

3. for 0<t<30, when does the reservoir have the most water? When does it have the least water?

4. What are the domain and range of r?

[stapel_eliz]

1. Let t be in seconds and let r(t) be the rate< in gallons/second, at which water enters a reservoir. We are given that, for t 0<t<30,: r(t) = 800 - 40t

1. evaluate the expressions r(0), r(15), r(25) and explain their physical significance.

Plug the given t-values into the given formula, and simplify to find the r-values.

Keeping in mind that "t" is defined as the time in seconds and "r" is defined as the rate of flow in gallons per second, state the r- and t-value pairs in terms of rate of flow r at a given time t. For instance, the value of r(5) would be the rate of flow, in gallons per second, at a time of five seconds.

2. if you were to graph y=r(t) for 0<t<30, labeling the intercepts. What is the physical significance of the slope and the intercepts?

To learn about the meaning of slope and y-intercept in the context of word problems, try here.

The x-intercept (well, okay; the t-intercept) is of course when r (usually y) equals zero. What does "r" stand for? What does "t" stand for? Interpret this intercept in terms of the rate of flow at that time t.

3. for 0<t<30, when does the reservoir have the most water? When does it have the least water?

Look at the graph. When (at what t-value) is the line the highest? When is it the lowest?

4. What are the domain and range of r?

They gave you the domain when they gave you the restrictions on t. To find the range, find the output values over the interval for which the function is defined.

Yes, "y = 800 - 40x" is defined "everywhere", but that isn't the domain they gave you. And yes, "y = 800 - 40x" goes "everywhere" (eventually attains every possible y-value), but that won't happen over the domain they gave you. So use the highest and lowest values from part (3) to state the range for r(t).

If you get stuck, please reply showing your work and reasoning so far. Thank you!

[santaclaus]

For 2. you told me to graph it by putting in the values of t, but I don't know how to solve for y because there are 2 variables y and r. y=r(t) when 0<t<30
for example if I put in 1 for t, y=r (1), so how do I solve it? Would it just be y=x? a line through zero and 1 being slope?
santaclaus wrote:
3. for 0<t<30, when does the reservoir have the most water? When does it have the least water?
Look at the graph. When (at what t-value) is the line the highest? When is it the lowest?
For this, do I graph r(t)=800=40t?
santaclaus wrote:
4. What are the domain and range of r?
They gave you the domain when they gave you the restrictions on t. To find the range, find the output values over the interval for which the function is defined. So, I understand the domain now, but for range, why do I take the answer from the last problem? do I just say all output values over interval for which the function is defined?
thanks for all your help - really helps.

[stapel_eliz]

The function is y = r(t). Plug in the t-values, and simplify to find r(t) = y. So, for instance, for t = 3, you would plug "3" in for "t", and simplify the formula to find the value for r, which you would graph as "y" (if you're using x,y-axes instead of t,r-axes).

To learn about function notation, try here.

Yes, to find the graph of r(t), you graph the function r, or, if you're more comfortable with the "usual" variables, graph y = 800 - 40x, so "x" stands for '"t" and "y" stands for "r".

The "range" is the output values, the y-values. The graph shows what you get for y-values after you plug in the x-values (or, technically, the r-values that you get after you plug in the t-values). The lesson in the link explains this further, and provides pictures.