91Ó°ÊÓ

In mathematics, parametric equations are a method of representing mathematical problems where one or more variables are expressed in terms of a third variable, often denoted as `t` for time. This approach can effectively define paths of motion, such as the position of a moving particle over time.
Parametric equations are vital for illustrating curves because they clearly describe both the x-coordinate and the y-coordinate as functions of one parameter. Therefore, instead of describing the path using a single equation like \( y = f(x) \), parametric equations can represent the motion in a format such as \( x(t) \) and \( y(t) \).
This exercise uses the concept where the particle's motion is directly described along the x-axis by a constant speed, simplifying the x-coordinate change to a function of time alone: \( x(t) = 3t \). The y-coordinate is determined from the curve equation \( y = \frac{2}{3}x^{3/2} \) by inserting the computed x-value at any time \( t \).

• Benefits: Represents both direction and dimension effectively.
• Can model real-world problems involving paths and trajectories.
• Makes complex curve descriptions simpler via parameterization.

Curves in the plane, in a mathematical sense, are continuous collections of points that extend without sharp angles or breaks. Each point on the curve corresponds to specific coordinates \((x, y)\) in a Cartesian plane. In this exercise, the particle flows along a curve defined by the equation \( y = \frac{2}{3} x^{3/2} \).
This particular curve is not a straight line but a smooth, gradually increasing path as determined by the power \( x^{3/2} \). Understanding curves helps visualize and analyze how objects move smoothly in relation to both axes in a plane.

• Application: Curves provide insight into trajectory and path in two dimensions.
• Understanding curves aids in predicting motion and smoothing paths.
• They offer intuitive understanding of objects under constant or variable velocities.

Distance and speed are two fundamental measures of motion in physics and mathematics. The speed is a scalar quantity measuring the rate of motion over time, while distance is the overall path length traveled independently of the path's direction.
In this exercise, the particle's speed is a constant 3 cm/sec, indicating it covers 3 cm of distance longitudinally along the x-axis every second. Over a time \( t = 6 \) seconds, the distance traveled is calculated by the formula: \( \text{distance} = \text{speed} \times \text{time} \), resulting in \( 18 \) cm.
This calculation remains linear as the particle travels uniformly in a straight horizontal direction at a steady pace; thus, there isn’t a need to resolve for different factors affecting motion based on variable speed or acceleration.

• Useful to quantify movement and understand dynamics of motion.
• Basic for determining trajectory and position in time-dependent scenarios.
• Essential for converting speed into spatial displacement over time.

Square roots often appear in mathematical equations related to geometry, motion, and measurement due to their principle of reverse-exponentiation, particularly converting a squared quantity back to its base value.
In this exercise, the function \( y = \frac{2}{3} x^{3/2} \) requires calculation of the square root portion \( x^{3/2} \), where the exponent fraction \( 3/2 \) denotes raising \( x \) first to the power of 1.5, equivalent to \( (\sqrt{x})^3 \).
When calculating for \( x=18 \), \( \sqrt{18} = 3\sqrt{2} \). Thus, to find \( (\sqrt{18})^3 \), it expands into the factored expression \( 27 \) times \( 2\sqrt{2} \), which simplifies some geometric curve points.

• Helpful for solving quadratic equations and polynomial roots.
• Widely used in rate analysis and scientific computations.
• Integral for simplifying and resolving radical expressions in mathematics.