Question

Find real numbers a and b such that equations \(A\) and \(B\) describe the same curve. $$\begin{aligned}&A: x=t+t^{3}, y=3+t^{2} ;-2 \leq t \leq 2\\\&\vec{B}: x=t^{1 / 3}+t, y=3+t^{2 / 3} ; a \leq t\leq b\end{aligned}$$

Short answer

Question: Determine the real numbers a and b, which define the valid range for t in equation B, such that both equations A and B describe the same curve. Answer: a = -1 and b = 1

Step-by-step solution

Compare equations

Begin by comparing x and y equations in both A and B. Let's analyze the relationship between the equations: For A: \(x = t + t^3\) \(y = 3 + t^2\) For B: \(x = t^{\frac{1}{3}} + t\) \(y = 3 + t^{\frac{2}{3}}\) In both A and B, y always starts with 3. For A, y has a quadratic term, while for B, y has a \(\frac{2}{3}\) power term. For x, in both A and B, there's a term with t, and then there's a term with the power of t. For A, it's a cubic term, and for B, it's a \(\frac{1}{3}\) power term.

Determine range for t in equation B

We are given the range for t in equation A, -2 ≤ t ≤ 2. We can use this range to determine the appropriate range for t in equation B so that the two curves described by the equations A and B match. Since in both A and B, y always starts with 3, let's compare the other terms: \(t^2 = t^{\frac{2}{3}}\) We need the range of t in B that satisfies the above equation within the given range of t in A. Raise both sides to the power of \(\frac{3}{2}\): \((t^2)^{\frac{3}{2}} = (t^{\frac{2}{3}})^{\frac{3}{2}}\) Simplify: \(t^3 = t\) Dividing both sides by t, we get: \(t^2 = 1\) So, \(t = -1\) or \(t = 1\) Since -2 ≤ t ≤ 2 for A, a suitable range for t in B can be obtained by taking into account both solution values, a = -1 and b = 1 which gives us the range -1 ≤ t ≤ 1. So, a = -1 and b = 1. The real numbers a and b are -1 and 1, respectively, such that equations A and B describe the same curve.

Key concepts

Curves

Curves in mathematics can be visualized as smooth and continuous lines that are described by equations in two dimensions. When we describe these curves using parametric equations, each point on the curve is given by the equations involving a third variable, often denoted as \(t\).
In this exercise, we have two parametric equations, A and B, which represent curves in the coordinate plane. By keeping track of how \(x\) and \(y\) change with respect to \(t\), we can sketch or understand the shape of these curves.
The parametric form allows us to explore complex shapes that might not be possible to express as a single function \(y = f(x)\). It’s a great way to analyze motion, where \(t\) can signify time or another varying quantity. This can help students visualize how changes in \(t\) result in changes along the curve, creating a path or trajectory.

Equation Comparison

When comparing parametric equations, you're looking for a way to see if the characteristics and parameters of the curves overlap or match.
To determine if two parametric equations describe the same curve, we must examine both \(x\) and \(y\) equations closely. For a given curve, if the structure of \(x\) and \(y\) for one parametric set maps completely onto another set by varying \(t\), the two can describe the same geometric path.
- In Equation A: - \(x = t + t^3\) - \(y = 3 + t^2\)- In Equation B: - \(x = t^{1/3} + t\) - \(y = 3 + t^{2/3}\)Here, the two equations are compared to check the power terms of \(t\). They are both constructed using a combination of \(t\) terms with different exponents, affecting the shape and transformation of the curve they describe.
Understanding these equations allows us to establish a connection by adjusting the parameter range to match the curvature stipulated by each equation.

Range Determination

Determining the appropriate range of the parameter \(t\) is crucial to ensuring that the curves described by the equations match up perfectly. For parametric equations, adjusting the range of \(t\) can alter the section of the curve you are focusing on.
For Equation A, we have the range \(-2 \leq t \leq 2\). This range determines how \(t\) varies to describe the entire curve, ensuring all unique points on the curve are covered. When trying to make Equation B describe the same curve, we encounter a different set of expressions for \(x\) and \(y\).
To find the proper range for Equation B:- Reconcile differences in the equations by matching respective powers of \(t\).- Solve the equation \(t^3 = t\) for joint values of \(t\).- The solutions \(t = -1\) and \(t = 1\) inform us that the range \(-1 \leq t \leq 1\) for B will delineate the same portion of the curve as Equation A.
By adjusting the parameter range correctly, you can superimpose one curve onto another, effectively equating the geometric paths described by different parametric definitions.