Question

Determine the domain that results from eliminating the parameter in the set of parametric equations below. x(t) = 6t+3 y(t) = 7√t+3 Enter your answer using interval notation.

Answer 1

The domain resulting from eliminating the parameter in the set of parametric equations (x(t) = 6t + 3) and
`\(y(t) = 7√(t) + 3\)` is
`\([3, +\infty)\)`.

Step-by-step explanation:

The elimination of the parameter involves finding the possible values of (t) that allow the parametric equations to define valid points in the xy-plane. In this case, the square root term in the equation
`\(y(t) = 7√(t) + 3\)` requires (t) to be non-negative to avoid imaginary numbers. Thus,
`\(t \geq 0\)`.

Next, since the domain represents the possible values for \(x\), we consider the expression (x(t) = 6t + 3). As (t) is already restricted to be non-negative, the minimum value of (x) occurs when (t = 0), yielding (x(0) = 3). Therefore, the domain is
`\([3, +\infty)\)` as (x) increases without bound as (t) increases.

In summary, the domain resulting from eliminating the parameter is
`\([3, +\infty)\)`, signifying that the parametric equations define points along or to the right of the vertical line (x = 3) in the xy-plane. The initial restriction on (t) ensures that the square root term in the (y) equation is always real, and the domain reflects the continuous, increasing nature of the (x) parameter.

Answer 2

To find the domain of the given parametric equations after eliminating the parameter, substitute the expressions for x(t) and y(t) into the equation y(t) and solve for y(x). The resulting equation will have a domain of x ≥ 3.

Step-by-step explanation:

The given parametric equations are x(t) = 6t + 3 and y(t) = 7√(t + 3).

To eliminate the parameter t and find the domain, we need to solve for t in terms of x or y.

From the equation x(t) = 6t + 3, we can isolate t: t = (x - 3)/6.

Substitute this expression into the equation y(t) = 7√(t + 3) to get y(x): y(x) = 7√(((x - 3)/6) + 3).

The domain of the resulting equation is the set of all x-values that make the expression inside the square root non-negative. Solving this inequality, we get x ≥ 3.

Therefore, the domain of the parametric equations, after eliminating the parameter, is [3, ∞).