To find the arc length of a curve given by parametric equations $x = 2t + 5$ and $y = 7 - 7t$ on the interval $-1 \leq t \leq 3$, we use the formula for arc length of a parametric curve:

$L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt$

Step 1: Differentiate $x$ and $y$ with respect to $t$.

For $x = 2t + 5$: $\frac{dx}{dt} = 2$

For $y = 7 - 7t$: $\frac{dy}{dt} = -7$

Step 2: Substitute into the arc length formula.

$L = \int_{-1}^{3} \sqrt{(2)^2 + (-7)^2} \, dt$

Simplify inside the square root: $L = \int_{-1}^{3} \sqrt{4 + 49} \, dt = \int_{-1}^{3} \sqrt{53} \, dt$

Since $\sqrt{53}$ is a constant, we can factor it out of the integral: $L = \sqrt{53} \int_{-1}^{3} \, dt$

Step 3: Evaluate the integral.

$L = \sqrt{53} \cdot [t]_{-1}^{3} = \sqrt{53} \cdot (3 - (-1)) = \sqrt{53} \cdot 4$

Step 4: Compute the final result.

$L = 4 \sqrt{53} \approx 4 \times 7.280 = 29.121$

Final Answer

The arc length is approximately: $L \approx 29.121$

Would you like further clarification on any of the steps?

Here are five related questions that might deepen your understanding:

1. How does the arc length formula change if we have a curve defined in polar coordinates?
2. How would we find the arc length if we were given functions $y = f(x)$ instead of parametric equations?
3. What happens to the arc length formula if the interval is a closed loop?
4. How would changing the interval to $-1 \leq t \leq 2$ affect the arc length calculation?
5. How would the process change if we had a 3-dimensional parametric curve with $z = g(t)$ as well?

Tip: Always simplify square roots before evaluating integrals when possible.