Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Please solve the problem below. Thank you!
![### Finding the Tangent Line to a Parametric Curve
**Problem:**
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.
Given:
\[ x = t \cos(t), \quad y = t \sin(t); \quad t = \pi \]
**Solution:**
To find the equation of the tangent line, we need to determine the slope of the curve at \( t = \pi \) and then use the point-slope form of the equation of a line.
1. **Find the coordinates of the point on the curve at \( t = \pi \):**
\[
x(\pi) = \pi \cos(\pi) = \pi \cdot (-1) = -\pi
\]
\[
y(\pi) = \pi \sin(\pi) = \pi \cdot 0 = 0
\]
So, the point is \( (-\pi, 0) \).
2. **Find the derivatives \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \):**
\[
\frac{dx}{dt} = \cos(t) - t \sin(t)
\]
\[
\frac{dy}{dt} = \sin(t) + t \cos(t)
\]
3. **Evaluate the derivatives at \( t = \pi \):**
\[
\left. \frac{dx}{dt} \right|_{t = \pi} = \cos(\pi) - \pi \sin(\pi) = -1 - 0 = -1
\]
\[
\left. \frac{dy}{dt} \right|_{t = \pi} = \sin(\pi) + \pi \cos(\pi) = 0 - \pi = -\pi
\]
4. **Find the slope of the tangent line:**
The slope \( m \) of the tangent line is given by:
\[
m = \frac{dy/dt}{dx/dt} = \left. \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \right|_{t = \pi} = \frac{-\pi}{-1} = \pi
\]
5. **Use the point](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd1148f2e-12e2-4d7a-aa19-e54736499ad8%2Fa5521ff5-32e0-4bc7-a5ae-3b2431397c13%2Fivk837l_processed.png&w=3840&q=75)
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