Given the following position function:
(position is in ft , time is in sec )

Find the velocity at .

Round your answer to the nearest thousandth.

Ask by Gibson Phillips. in the United States

Mar 29,2025

Question

Given the following position function:
(position is in ft , time is in sec )

Find the velocity at .

Round your answer to the nearest thousandth.

Ask by Gibson Phillips. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

We are given the position function  
$$
s(t)=\frac{4t+5}{t+1}.
$$

The velocity is the derivative of the position function with respect to time, $$ s'(t) $$.

**Step 1. Differentiate using the Quotient Rule**

The quotient rule states that if  
$$
f(t)=\frac{g(t)}{h(t)},
$$  
then  
$$
f'(t)=\frac{g'(t)h(t)-g(t)h'(t)}{\left[h(t)\right]^2}.
$$

For our function, let  
$$
g(t)=4t+5 \quad \text{and} \quad h(t)=t+1.
$$

Differentiate both:
$$
g'(t)=4 \quad \text{and} \quad h'(t)=1.
$$

Now apply the quotient rule:
$$
s'(t)=\frac{(4)(t+1)-(4t+5)(1)}{(t+1)^2}.
$$

**Step 2. Simplify the Expression**

Expand the numerator:
$$
4t+4 - 4t - 5.
$$

Simplify:
$$
4t+4-4t-5 = -1.
$$

Thus, the derivative becomes:
$$
s'(t)=\frac{-1}{(t+1)^2}.
$$

**Step 3. Evaluate the Derivative at $$ t=2 $$**

Plug in $$ t=2 $$:
$$
s'(2)=\frac{-1}{(2+1)^2}=\frac{-1}{3^2}=\frac{-1}{9}.
$$

**Step 4. Round the Answer to the Nearest Thousandth**

Compute the decimal value:
$$
\frac{-1}{9}\approx -0.111.
$$

Thus, the velocity at $$ t=2 $$ is:
$$
\mathrm{v}(2)=-0.111\frac{\text{ft}}{\text{sec}}.
$$
The velocity at $$ t=2 $$ is $$ -0.111 \frac{\text{ft}}{\text{sec}} $$.

Given the following position function:
(position is in ft , time is in sec )

Find the velocity at .

Round your answer to the nearest thousandth.

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Given the following position function: (position is in ft , time is in sec ) Find the velocity at . Round your answer to the nearest thousandth.

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Given the following position function:
(position is in ft , time is in sec )

Find the velocity at .

Round your answer to the nearest thousandth.