QuestionQuestion
A particle has a velocity modeled by , measured in , over the interval . What is distance traveled from to ?
(You may either enter an exact answer or round your answer to the nearest tenth of a centimeter.)
Provide your answer below:
cm
Studdy Solution
STEP 1
What is this asking?
How far does a particle move in 7 seconds if its speed changes over time according to a logarithmic formula?
Watch out!
Velocity can be negative, but distance is always positive!
We need the *absolute value* of the velocity if we're going backwards.
Luckily, in this problem, the velocity is always positive.
STEP 2
1. Set up the distance integral
2. Solve the integral
3. Calculate the result
STEP 3
Alright, so we're given the velocity function , and we want to find the *total distance* traveled between and .
Remember, distance is just the integral of the *absolute value* of velocity.
STEP 4
In our case, is always positive between and , so the absolute value doesn't change anything here.
Phew! That makes our lives easier.
The distance traveled is given by:
STEP 5
Let's **change the base** of the logarithm from base 4 to base *e* using the **change of base formula**: .
STEP 6
We can **pull the constants** out of the integral:
STEP 7
Now, we can use **integration by parts** to solve the integral of .
Let and .
Then and .
STEP 8
The integration by parts formula is , so:
STEP 9
To solve , we can **add zero** in the numerator: .
STEP 10
Putting it all together:
STEP 11
So our **definite integral** becomes:
STEP 12
Let's **plug in the limits of integration**:
STEP 13
We can simplify to :
STEP 14
Calculating this gives us approximately:
STEP 15
The distance traveled is approximately **19.96 cm**.
Was this helpful?