Find 3x+27=12+14x-112

Mathematics

1 answer:

-Dominant- [34]4 years ago

3 0

Answer:

$\large\boxed{x=\dfrac{127}{11}}$

Step-by-step explanation:

$3x+27=12+14x-112\\\\3x+27=14x+(12-112)\\\\3x+27=14x-100\qquad\text{subtract 27 from both sides}\\\\3x+27-27=14x-100-27\\\\3x=14x-127\qquad\text{subtract }\ 14x\ \text{from both sides}\\\\3x-14x=12x-12x-127\\\\-11x=-127\qquad\text{divide both sides by (-11)}\\\\\dfrac{-11x}{-11}=\dfrac{-127}{-11}\\\\x=\dfrac{127}{11}$

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What is the length of the curve with parametric equations x = t - cos(t), y = 1 - sin(t) from t = 0 to t = π? (5 points)

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General Formulas and Concepts:

Calculus

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Derivatives
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Basic Power Rule:

f(x) = cxⁿ
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Arc Length Formula [Parametric]: $\displaystyle AL = \int\limits^b_a {\sqrt{[x'(t)]^2 + [y(t)]^2}} \, dx$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \left \{ {{x = t - cos(t)} \atop {y = 1 - sin(t)}} \right.$

Interval [0, π]

Step 2: Find Arc Length

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Substitute in variables [Arc Length Formula - Parametric]: $\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{[1 + sin(t)]^2 + [-cos(t)]^2}} \, dx$
[Integrand] Simplify: $\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{2[sin(x) + 1]} \, dx$
[Integral] Evaluate: $\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{2[sin(x) + 1]} \, dx = 4\sqrt{2}$