Your velocity is given by $$ v(t)=t^{2}+1 \mathrm{in} \mathrm{m} / \mathrm{sec} $$, with $$ t $$ in seconds. Estimate the distance, $$ s $$, traveled between $$ t=0 $$ and $$ t=5 $$. Use an overestimate with data every one second. The distance is approximately i eTextbook and Media

Question

UpStudy AI · Accepted Answer

To estimate the distance traveled between $$ t = 0 $$ and $$ t = 5 $$ seconds using an overestimate with data every one second, we can use the velocity function $$ v(t) = t^2 + 1 $$ to calculate the velocity at each integer second and then sum the areas of rectangles formed by these velocities.

1. **Calculate the velocity at each second:**
   - $$ v(0) = 0^2 + 1 = 1 \, \text{m/s} $$
   - $$ v(1) = 1^2 + 1 = 2 \, \text{m/s} $$
   - $$ v(2) = 2^2 + 1 = 5 \, \text{m/s} $$
   - $$ v(3) = 3^2 + 1 = 10 \, \text{m/s} $$
   - $$ v(4) = 4^2 + 1 = 17 \, \text{m/s} $$
   - $$ v(5) = 5^2 + 1 = 26 \, \text{m/s} $$

2. **Use the velocities to estimate the distance:**
   Since we want an overestimate, we will use the velocity at the end of each interval to calculate the distance for that interval. The distance traveled in each second is given by the velocity at the end of that second multiplied by the time interval (which is 1 second).

- Distance from $$ t = 0 $$ to $$ t = 1 $$: $$ v(1) \cdot 1 = 2 \cdot 1 = 2 \, \text{m} $$
   - Distance from $$ t = 1 $$ to $$ t = 2 $$: $$ v(2) \cdot 1 = 5 \cdot 1 = 5 \, \text{m} $$
   - Distance from $$ t = 2 $$ to $$ t = 3 $$: $$ v(3) \cdot 1 = 10 \cdot 1 = 10 \, \text{m} $$
   - Distance from $$ t = 3 $$ to $$ t = 4 $$: $$ v(4) \cdot 1 = 17 \cdot 1 = 17 \, \text{m} $$
   - Distance from $$ t = 4 $$ to $$ t = 5 $$: $$ v(5) \cdot 1 = 26 \cdot 1 = 26 \, \text{m} $$

3. **Sum the distances:**
   $$
   s \approx 2 + 5 + 10 + 17 + 26 = 60 \, \text{m}
   $$

Thus, the estimated distance traveled between $$ t = 0 $$ and $$ t = 5 $$ seconds is approximately $$ \boxed{60} $$ meters.
The estimated distance traveled between $$ t = 0 $$ and $$ t = 5 $$ seconds is approximately 60 meters.

Your velocity is given by , with in seconds.
Estimate the distance, , traveled between and .
Use an overestimate with data every one second.
The distance is approximately i
eTextbook and Media

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

Your velocity is given by , with in seconds. Estimate the distance, , traveled between and . Use an overestimate with data every one second. The distance is approximately i eTextbook and Media