All categories

3 months ago

For the given position vectors r(t) compute the unit tangent vector T(t) for the given value of t .

Mathematics

1 answer:

Svet_ta [12.7K]3 months ago

5 0

Answer:

a) $T(t) = \frac{}{5}=$

$T(4) =$

b) $T(t) = \frac{}{8\sqrt{37}}$

$T(4) =$

c) $T(t) = \frac{}{2425825977}$

$T(4) = \frac{1}{2425825977}$

Step-by-step explanation:

The tangent vector is determined as follows:

$T(t) = \frac{r'(t)}{|r'(t)|}$

Part a

The given function for this instance is:

$r(t) =$

The derivative is calculated as:

$r'(t) =$

The magnitude of the derivative is determined by:

$|r'(t)| = \sqrt{25 sin^2(5t) +25 cos^2 (5t)}= 5\sqrt{cos^2 (5t) + sin^2 (5t)} =5$

Thus, the tangent vector is found to be:

$T(t) = \frac{}{5}=$

Finally, for t=4 we find:

$T(4) =$

Part b

In this case, we have the stated function:

$r(t) =$

The derivative can be expressed as:

$r'(t) =$

The magnitude for the derivative is calculated as:

$|r'(t)| = \sqrt{4t^2 +9t^4}= t\sqrt{4 + 9t^2}$

$|r'(4)| = \sqrt{4(4)^2 +9(4)^4}= 4\sqrt{4 + 9(4)^2} = 4\sqrt{148}= 8\sqrt{37}$

Thus, the tangent vector in this case is:

$T(t) = \frac{}{8\sqrt{37}}$

Finally, when t=4 we find:

$T(4) =$

Part c

Here, we have the corresponding function stated:

$r(t) =$

The derivative is stated as:

$r'(t) =$

The magnitude for the derived value is calculated as:

$|r'(t)| = \sqrt{25e^{10t} +16e^{-8t} +1}$

$|r'(t)| = \sqrt{25e^{10*4} +16e^{-8*4} +1} =2425825977$

Therefore, we conclude the tangent vector here:

$T(t) = \frac{}{2425825977}$

And for the case when t=4, we get:

$T(4) = \frac{1}{2425825977}$

You might be interested in

Yolanda is buying supplies for school. She buys n package of pencils at $1.40 per package and m pads of paper at $1.20 each. Wha

Inessa [12570]

To tackle this issue, we need to clarify what each variable stands for. Here, n indicates the number of pencil packages purchased by Yolanda, while m represents the number of paper pads she buys. We calculate the total cost by multiplying these variables by their individual prices, which helps us determine the expense for both pencil packages and pads of paper.

The expression "1.4n" indicates the overall cost for n pencil packages.

On the other hand, "1.2m" signifies the complete expense for m pads of paper.

As we sum these two expressions, the resulting total reflects the overall expenditure for both items.

The equation "1.4n + 1.2m" summarizes this calculation.

I hope this provides clarity!

8 0

4 months ago

Tran has a credit card with a spending limit of $2000 and an APR (annual percentage rate) of 12%. During the first month, Tran c

babunello [11817]

Detailed explanation:

Information provided:

Tran possesses a credit card that allows up to $2000 in spending with an APR of 12%.

In the initial month, Tran incurred charges of $450 and settled $150 within that billing period.

The formula to determine the interest that will accrue for Tran in the first month is (0.012)(300)

Here, 0.01 signifies the monthly interest rate.

The 300 reflects the outstanding balance, as Tran charged $450 but only paid back $150.

5 0

3 months ago

Wendy is arranging books on the bookshelves in the school library. The total number of books she arranges is given by the equati

Zina [12379]

This should be straightforward; are you even a high school student? Just kidding! The calculation is 12*16 since there are 16 rows, and r indicates the rows, resulting in a total of 192 books.<==== Answer.

4 0

3 months ago

Read 2 more answers