MYTUTOR SUBJECT ANSWERS

282 views

A cup of coffee is cooling down in a room following the equation x = 15 + 70e^(-t/40). Find the rate at which the temperature is decreasing when the coffee cools to 60°C.

First, we need to find the value of t when x = 6p°C. We are told that after t minutes the temperature, x, will be 60°C; so we can insert 60 into the equation for x: 

60 = 15 + 70e^(-t/40)

Secondly, we can rearrange the equation to get like terms on each side, meaning we subtract 15 from both sides. 

60 - 15 = 15 + 70e^(-t/40) - 15

45 = 70e^(-t/40)

Thirdly, we can divide both sides by 70 to get the 'e' term on its own. This will make the final step for this part of the question easier, but isn't necessarily needed at this stage:

45/70 = (70e^(-t/40))/70

Simplify: 9/14 = e^(-t/40)

Fourthly, take the ln of both sides to remove the e function, and divide by -1/40 to isolate t:

ln(9/14) = ln(e^(-t/40))

ln(9/14) = -t/40

-40ln(9/14)  = t

t =~ 17.67 mins

To solve the second part of the question we first need to differentiate the initial equation:

x = 15 + 70e^(-t/40)

The differential of an exponential function is the first derivative of the term the function is applied to, -t/40. Differentiating this with respect to t is simply -1/40. Remembering the product rule tells us to multiply this by the initial 70:

dx/dt = -70/40e^(-t/40)

Simplify: dx/dt = -7/4e^(-t/40)

Finally, substitute the saved value of t into this equation:

dx/dt = -7/4e^(-(-40ln(9/14))/40)

        = -9/8°C/min

Therefore the temperature is decreasing at 9/8°C/min. Remember the question asks for the rate of decrease so the answer should be positive. You may lose marks if you leave the answer negative. 

Nathan A. IB Maths tutor, A Level Maths tutor, 13 plus  Maths tutor, ...

4 months ago

Answered by Nathan, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

181 SUBJECT SPECIALISTS

£22 /hr

Kirsty S.

Degree: Mathematics (Bachelors) - Warwick University

Subjects offered: Maths, Spanish+ 1 more

Maths
Spanish
Further Mathematics

“Hello, I'm Kirsty and I am here to help you with Maths. Maths will soon become the exam that you will look forward to, so you can show off how much you know! Before I begin any tuition I find out exactly what you would like to get ou...”

£20 /hr

Georgiana P.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered: Maths

Maths

“I am a Maths student at the University of Bristol and hope to share my love of maths with you. I believe that Maths is learnt in 3 steps:    1) I will explain and we will discuss the main principles of what we are working on, assis...”

£20 /hr

Shandon W.

Degree: Combined Honours in Social Sciences (Economics, Spanish) (Bachelors) - Durham University

Subjects offered: Maths, Spanish+ 2 more

Maths
Spanish
Economics
Chemistry

“I am currently a second year student at the University of Durham, studying Economics and Spanish. I have always found it easy to take on difficult concepts, and I hope to be able to pass on this understanding to other students. Having...”

MyTutor guarantee

About the author

£20 /hr

Nathan A.

Degree: Accounting and Finannce (Bachelors) - Bristol University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Economics

“Hi there! My name is Nathan Amoah and I am a second year Accounting and Finance student at the University of Bristol. I have always had a passion for finding ways to reduce problems to their most basic level; and this is a philosphy t...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

What methods are there for integration?

Find the exact gradient of the curve y=ln(1-cos2x) at the point with x-coordinate π/6

Below is a question from the Edexcel Maths Core 1 textbook, Solve the equation x^2 + 8x + 10 = 0 using completing the square.

Explain briefly the Normal Distribution

View A Level Maths tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok