Currently unavailable: for new students
Degree: Engineering Science (Masters) - Oxford, Keble College University
|Maths||A Level||£24 /hr|
|Physics||A Level||£24 /hr|
|Maths||13 Plus||£22 /hr|
|Science||13 Plus||£22 /hr|
|Maths||11 Plus||£22 /hr|
|-Oxbridge Preparation-||Mentoring||£24 /hr|
|-Personal Statements-||Mentoring||£24 /hr|
|Engineering Science||Masters Degree||2:2|
|Before 12pm||12pm - 5pm||After 5pm|
Please get in touch for more detailed availability
Jacob (Parent) October 14 2016
Manon (Student) October 13 2016
Pierre (Student) October 8 2016
Jacob (Parent) December 12 2016
(The rate at which the temperature of a body falls is proportional to the difference between the body temperature and the room temperature) - this would be given in the question.
This is a typical ordinary differential equation (ODE) question, which can be solved by separation of variables.
It is best looked at in three stages: Write the general ODE, then find the boundary conditons, then solve and find the equation for temperature in terms of time.
1. General ODE: let's call temperature T, and time t. Change is temperature over time is written as dT/dt. From the question, dT/dt is proportional to (T-18).
So dT/dt = -k(T-18) where k is the constant of proportionality and the minus sign arrises because the pan is cooling not heating.
2. Boundary conditions:
The pan of water is heated to 100C which is at t =0, so T(t=0) = 100.
After 5 minutes, the pan cools to 80C. So T(t=5) = 80
3. Solve the equation and substitute in the boundary conditions.
Firstly, we separate the variables and integrate.
dT/dt = -k(T-18)
dT = -k(T-18) dt
dT/(T-18) = -k dt
Add the integral signs (int):
int dT/(T-18) = int -k dt
ln(T-18) = -kt + C where C is the constant of integration.
Now do exp of both sides. exp is a way of wirting e to the power of ( ):
T - 18 = exp(-kt + C)
T - 18 = A exp(-kt) where A = exp(C). This is just used to simplify the problem.
T = 18 + A exp(-kt)
Seconly, we substitute in the boundary conditions. Use T(t=0) = 100 to start with.
100 = 18 + Aexp(0)
so A = 82
T = 18 + 82 exp(-kt)
Now it's time to use the 2nd boundary condition, T(t=5) = 80.
80 = 18 + 82 exp(-5k)
Subtract 18 from both sides, and divide by 82:
(80 - 18)/82 = exp(-5k)
Take ln of both sides and then divide by -5
k = -(1/5) ln(82/62)
k = 0.0559
So the final solution is:
T = 18 + 82 exp(-0.0559t)
We now have the general solution, which can be used to find any further information about the quesiton we need.
The question asks for the temperature at t=10. Simply substitute t-10 into the final solution above.
T = 18 + 82 exp(-0.0559 x 10)
T = 64.0C (three sig fig)see more
Solve for x: 2x2 + 7x + 6 = 0
The first thing to do is look at the left hand side and factorise it. To do this we use the reverse of FOIL. Then when the expression is factorised, we can solve for x.
1. First find the skeleton with just x's:
(2x )(x )
2. Next, find 2 numbers that mutiply to make 6, and these go at the end of the brackets. This could be 6 and 1 or 2 and 3, or -6 and -1 or -2 and -3.
Since there is a 7x term, the 2 numbers must add to 7 when one of them is multiplied by 2 - due to the 2x term (2x ).
so (2x 3)(x 2) or (2x 2)( 3)
3. Trial and error shows us that (2x + 3)(x + 2) is the right answer.
4. Now set each bracket respectively to 0 and solve for x. The solutions are x = -3/2 and x = -2.
And there we have the answers!see more