Exponential growth: water depth modeled by d = 4.2t + 22 (0<=t<=20)

[cdot]

I am having a nightmare with this question. If anyone can help me it would be appreciated.

So here goes.

The depth of water in a tank over a 20-minute time period can be
modelled by the equation

d = 4.2t + 22 (0<=t<=20)

where d is the depth of water in centimetres, and t is the time in
minutes after the water begins to flow.

(i) Find the depth of water in the tank 15 minutes after the water
begins to flow.

(ii) Calculate the time at which the depth of water is 80 centimetres.

(iii) Write down the gradient of the straight line represented by the
equation d = 4.2t + 22. What does this represent in the practical
situation being modelled?

(iv) Write down the value of the vertical (d) intercept. Interpret this in
the context of the practical situation being modelled.

(v) Either using Graphplotter or by hand, sketch the graph of
d = 4.2t + 22, putting d on the vertical axis and covering the time
interval 0<=t<=20.

 

[stapel]

The depth of water in a tank over a 20-minute time period can be modelled by the following equation:

. . . . .d = 4.2t + 22, for 0 < t < 20

...where d is the depth of water in centimetres, and t is the time in minutes after the water begins to flow.

This is a linear equation. On what basis are you thinking that this is "exponential" growth, rather than "linear" growth?

(i) Find the depth of water in the tank 15 minutes after the water begins to flow.

You are given a formula for the depth d after t minutes of water flow. You are given a value for the minutes of water flow. Plug the given number into the given formula for the specified variable, and simplify to find the specified value.

(ii) Calculate the time at which the depth of water is 80 centimetres.

You are given a formula for the depth d after t minutes of water flow. You are given a value for depth. Plug the given number into the given formula for the specified variable, and simplify to find the specified value.

(iii) Write down the gradient of the straight line represented by the equation d = 4.2t + 22. What does this represent in the practical situation being modelled?

(iv) Write down the value of the vertical (d) intercept. Interpret this in the context of the practical situation being modelled.

To learn how to interpret linear equations in terms of what the slope and y-intercept indicate, try here.

(v) Either using Graphplotter or by hand, sketch the graph of d = 4:2t + 22, putting d on the vertical axis and covering the time interval 0 < t < 20.

To learn how to graph straight-line equations, try here.

If you get stuck, please reply showing your work (even if you think it's wrong) for each of the parts above. Thank you!

 

[cdot]

This is a linear equation. On what basis are you thinking that this is "exponential" growth, rather than "linear" growth?

I just thought it was exponential growth as the less than or equal part put me off.

(i) Find the depth of water in the tank 15 minutes after the water begins to flow.

You are given a formula for the depth d after t minutes of water flow.

You are given a value for the minutes of water flow. Plug the given number into the given formula for the specified variable, and simplify to find the specified value.

I am still not sure what to do here, I can't wrap my head around how I pursue the formula, and where the question gets the data for the 20 minute time period already.

If i can get the answer to this one, i can surely do the rest on my own knowing the correct formula.

Thanks
Christoph

 

[mmm4444bot]

I am still not sure what to do [for part (i)] … I can't wrap my head around how I pursue the formula …

You do not need to pursue any formula; they have given you the formula to use.

d = 4.2*t + 22

Part (i) provides a value of 15 for t, and it asks you to calculate the corresponding value of d.

Substitute 15 for t and simplify. That is, you need to multiply 15 by 4.2, then add 22.

What do you get?

… where the question gets the data for the 20 minute time period …

It doesn't matter where the formula came from or why this model is valid for only 20 minutes of water flow because they're not asking any questions about where or how.

If i can get the answer to [part (i)], i can surely do the rest on my own knowing the correct formula.

In this exercise, there is only one formula, and you already have it:

d = 4.2t + 22

Please show us your work, for part (i), and we can go from there. :cool:

PS: The statement 0 ≤ t ≤ 20 tells us that values of t are restricted to numbers from 0 through 20. In other words, their linear model for water depth is not valid for negative values of t or any value larger than 20.

 

[cdot]

In this exercise, there is only one formula, and you already have it:

d = 4.2t + 22

Please show us your work, for part (i), and we can go from there. :cool:

PS: The statement 0 ≤ t ≤ 20 tells us that values of t are restricted to numbers from 0 through 20. In other words, their linear model for water depth is not valid for negative values of t or any value larger than 20.

d=4.2*15+22=85cm in depth

this is what i get?

 

[cdot]

simplifying to 58/4.2 = 13.089

 

[Deleted member 4993]

simplifying to 58/4.2 = 13.089

Where did you get that number?

Please put reference to the part of the question you are answering.

 

[mmm4444bot]

d=4.2*15+22=85cm in depth

Yes, 85cm is the correct answer for Part (i).

d = 85 when t = 15

In context, it means the water depth measures 85cm, after water has flowed into the tank for 15 minutes.

 

[mmm4444bot]

simplifying to 58/4.2 = 13.089

This looks like the last step of Part (ii). As Subhotosh noted, it helps when students post all the steps (at least the first one), instead of just the result, so we don't have to hunt for the meaning.

Part (ii):

d = 4.2t + 22

80 = 4.2t + 22

80 - 22 = 4.2t

58 = 4.2t

t = 58/4.2

I think you mistyped your result above. Also, it's not properly rounded.

58/4.2 = 13.8095 (rounded to the nearest ten-thousandth of a minute)

The exercise does not specify how they want this answer rounded, so you're free to choose.

If I were doing the exercise, I would round to the nearest tenth of a minute.

d = 80 when t ≈ 13.8

In context, it means that the water depth measures 80cm, after water has been flowing into the tank for about 13 minutes and 48 seconds.

For Part (iii), can you identify the gradient, from the given formula? They ask you to explain the meaning of this number, in the context of the exercise. In your own words, how would you describe this number? :cool: