MYTUTOR SUBJECT ANSWERS

164 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, ...

3 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

183 SUBJECT SPECIALISTS

£20 /hr

Jack G.

Degree: Politics, Philosophy, and Economics (Bachelors) - Warwick University

Subjects offered: Maths, Religious Studies+ 2 more

Maths
Religious Studies
Economics
.TSA. Oxford.

“Hello, I'm Jack and am studying politics, philosophy, and economics at Warwick.  Having recently been through the subjects I'm looking to tutor I feel like I know first hand what difficulties you'll be facing. I always found I wanted ...”

MyTutor guarantee

£20 /hr

Hafsah K.

Degree: Mechanical Engineering (Masters) - Nottingham University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Chemistry

“About Me:  I study Mechanical Engineering at the University of Nottingham. Maths and science have always been my favourite subjects, and I hope to share my passion for them with you! I have hadprevious experience in tutoring with two ...”

£20 /hr

Solomon L.

Degree: Mathematics (Bachelors) - Leeds University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“Currently studying at Leeds university, ready to help you improve your grades in Maths and Chemistry GCSE! ”

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

How do I integrate log(x) or ln(x)?

How do I find the turning points of a curve?

How to differentiate using the Product Rule

Differentiate y^3 + 3y^2 + 5

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