767 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.

11 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

#### 365 SUBJECT SPECIALISTS

£30 /hr

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“Hi, I'm Joe, an experienced tutor and a University of Bristol graduate with a first class Degree in Mathematics.”

£22 /hr

Degree: Electrical and Electronic Engineering (Masters) - Imperial College London University

Subjects offered:Maths, Physics+ 1 more

Maths
Physics
Chemistry

“I am passionate, motivated, and always hungry for more knowledge. I can bring energy and life to the classroom, and engage the student.”

£26 /hr

Degree: PGCE Secondary Mathematics (Other) - Leeds University

Subjects offered:Maths, Further Mathematics

Maths
Further Mathematics

“I am currently completing 2 PGCEs in Leeds. I have always had a passion for maths and my objective is to help as many as possible reach their full potential.”

£20 /hr

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 would you solve (2x+16)/(x+6)(x+7) in partial fractions?

Prove that the derivative of tan(x) is sec^2(x).

Given that y = 4x^3 – 5/(x^2) , x not equal to 0, find in their simplest form (a) dy/dx, and (b) integral of y with respect to x.

How can I find the area under the graph of y = f(x) between x = a and x = b?

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this.