Find the arc length of the curve defined by the parametric equations [tex]x(t)=5 e^{2 t}[/tex] and [tex]y(t)=3 e^{3 t}[/tex] from [tex]0.7 \leq t \leq 1.5[/tex]. Round your answer to the nearest hundredth.
[tex]s \approx \square[/tex]
Answer (2)
To find the arc length for the parametric equations from t = 0.7 to t = 1.5 , we calculate the derivatives, set up the arc length integral, and solve it numerically. The approximate arc length is 258.83 .
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Find the derivatives: d t d x = 10 e 2 t and d t d y = 9 e 3 t .
Square the derivatives: ( d t d x ) 2 = 100 e 4 t and ( d t d y ) 2 = 81 e 6 t .
Set up the arc length integral: s = ∫ 0.7 1.5 100 e 4 t + 81 e 6 t d t .
Evaluate the integral numerically to get the arc length: s ≈ 258.83 .
Explanation
Problem Setup We are given the parametric equations x ( t ) = 5 e 2 t and y ( t ) = 3 e 3 t , and we want to find the arc length of the curve from t = 0.7 to t = 1.5 .
Finding Derivatives First, we need to find the derivatives of x ( t ) and y ( t ) with respect to t .
d t d x = 10 e 2 t and d t d y = 9 e 3 t
Squaring Derivatives Next, we square these derivatives:
( d t d x ) 2 = ( 10 e 2 t ) 2 = 100 e 4 t and ( d t d y ) 2 = ( 9 e 3 t ) 2 = 81 e 6 t
Setting up the Arc Length Integral Now, we set up the arc length integral:
s = ∫ 0.7 1.5 ( d t d x ) 2 + ( d t d y ) 2 d t = ∫ 0.7 1.5 100 e 4 t + 81 e 6 t d t
Evaluating the Integral We need to evaluate this definite integral. Since it's difficult to find an elementary antiderivative, we can use a numerical method to approximate the integral. Using a numerical method, we find that:
s ≈ 258.83
Final Answer Therefore, the arc length of the curve is approximately 258.83.
Examples
In manufacturing, calculating the arc length of a curved component is crucial for determining the amount of material needed. For instance, if you're producing a curved metal strip described by parametric equations, knowing the arc length helps you cut the right amount of metal, reducing waste and ensuring accurate dimensions. This is especially important in industries like aerospace or automotive, where precision is key.
