Integrate (x^2+1)/(x+1) dx

For a cold drink in a hotter room, we can say that the rate of change of temperature of the drink is proportional to the difference of temperature between the drink and the room.

We can model that in this way

$\frac{dT}{dt}=k*(T_r-T)$

If we rearrange and integrate

$\int\frac{dT}{(T-Tr)} =-k*\int dt\\\\ln(T-T_r)=-kt+C1\\\\T-T_r=Ce^{-kt}\\\\T=T_r+Ce^{-kt}$

We know that at time 0, the temperature of the drink was 52°F. Then we have:

$T=T_r+Ce^{-kt}\\\\52=72+Ce^0=72+C\\\\C=-20$

We also know that at t=2, T=55°F

$T=T_r+Ce^{-kt}\\\\55=72-20e^{-k*2}\\\\e^{-k*2}=(72-55)/20=0.85\\\\-2k=ln(0.85)=-0.1625\\\\k=0.08$

The model for the temperature of the drink can be written as

$T=72-20e^{-0.08t}$

7 0

3 years ago

Which functions are symmetric with respect to the y-axis? Check all that apply.

xenn [34]

Answer:

f(x) = |x|, f(x) = [x] + 6

Step-by-step explanation:

Almost all of these are absolute values equations, which means the y doesn't change if x is positive or negative. The first one is the parent form, which is the simplest equation of the absolute equation, so it's symmetric with respect to the y-axis. The second equation is translated 3 units to the left, and the third is translated 31 to the left. The forth is translated 6 up, so it's still symmetric with respect to the y-axis. The fifth is translated 61 units left, and the last one is simply a line, which isn't symmetric.

4 0

3 years ago

Find two consecutive even integers such that twice the smaller is 16 more than the larger

drek231 [11]

Define x:
Let one of the numbers be x.
The other number is x + 2

Construct equation:
twice the smaller is 16 more than the larger
⇒2x = x + 2 - 16

Solve x:
2x = x + 2 - 16
x = -14

Find the two numbers:
Smaller number = x = -14
Larger number = x + 2 = -12

Answer: The two numbers are -14 and -12

6 0

2 years ago

Read 2 more answers

Integrate (x^2+1)/(x+1) dx​

Integrate (x^2+1)/(x+1) dx