Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 41-42: As a particle moves along a straight path, its speed \(v\) (in \(\mathrm{cm} / \mathrm{sec}\) ) at time \(t\) (in seconds) is given by the equation. For what subintervals of the given time interval \([a, b]\) will its speed be at least \(k \mathrm{~cm} / \mathrm{sec}\) ? $$ v=t^{4}-4 t^{2}+10 ; \quad[1,6] ; \quad k=10 $$

Step-by-Step Solution

Answer

The speed is at least 10 cm/sec on \([2, 6]\).

Step 1: Understand the Problem

The problem asks us to find the time intervals where the speed of a particle is not less than 10 cm/sec. The speed is given by the function \( v(t) = t^4 - 4t^2 + 10 \) over the interval \([1, 6]\).

Step 2: Set up the Inequality

To find the intervals where speed is at least 10 cm/sec, set up the inequality \( v(t) \geq 10 \). For this particular function, \( t^4 - 4t^2 + 10 \geq 10 \).

Step 3: Simplify the Inequality

Simplify the inequality: \( t^4 - 4t^2 + 10 \geq 10 \) simplifies to \( t^4 - 4t^2 \geq 0 \).

Step 4: Factor the Expression

Factor \( t^4 - 4t^2 \) to \( t^2(t^2 - 4) \). Further simplify to \( t^2(t - 2)(t + 2) \).

Step 5: Solve for Critical Points

Set each factor equal to zero: \( t^2 = 0 \), \( t - 2 = 0 \), and \( t + 2 = 0 \). Solving these gives us critical points: \( t = 0, 2, -2 \). Since we are only interested in \([1, 6]\), only \( t = 2 \) is relevant.

Step 6: Test Intervals

Since \( t = 2 \) is the critical point in \([1, 6]\): test the intervals \([1, 2]\) and \([2, 6]\):- For \( t \) in \( [1, 2) \), choose \( t = 1.5 \). Calculate \( v(1.5) \).- For \( t \) in \( (2, 6] \), choose \( t = 4 \). Calculate \( v(4) \) and also verify edge values \( v(2) \), \( v(6) \).Calculate values:1. \( v(1.5) = (1.5)^4 - 4(1.5)^2 + 10 = 5.0625 \) (not \( \geq 10 \))2. \( v(4) = 4^4 - 4(4)^2 + 10 = 186 \) 3. \( v(2) = 2^4 - 4(2)^2 + 10 = 10 \)4. \( v(6) = 6^4 - 4(6)^2 + 10 = 1130 \).Wherefore, \( t\in (2, 6] \).

Step 7: State the Solution

The speed of the particle is at least 10 cm/sec between the interval \([2, 6]\).

Key Concepts

Polynomial Inequality

Polynomial inequalities are equations where you need to determine the range of values for a variable that will satisfy a given inequality condition.
The polynomial inequality in this exercise is given by the speed function of the particle: \( t^4 - 4t^2 + 10 \geq 10 \). Here, you want to find when the speed is at least 10 cm/sec.
When solving a polynomial inequality, start by simplifying it. Here, it is simplified by subtracting 10 from both sides leading to: \( t^4 - 4t^2 \geq 0 \). This simplified version makes it easier to identify critical points and analyze intervals, helping us understand when the inequality holds true.

Interval Testing

Interval testing is a method used to determine which parts of a domain satisfy a certain condition, especially after finding critical points of the function.
Once identified, these intervals must be tested to check where the original inequality holds.
For the example \( t^4 - 4t^2 \geq 0 \), testing is performed on intervals divided by the critical point \( t = 2 \) from the domain \([1, 6]\).
Interval testing involves picking test values from each subinterval, like \( t = 1.5 \) for \([1, 2)\), and \( t = 4 \) for \((2, 6]\), to evaluate if \( v(t) \geq 10 \).
This step ensures clarity in determining where the inequality holds within each suggested interval.

Critical Points

Critical points are values of \( t \) where the function changes from increasing to decreasing, or vice versa, or remains stationary. These are typically where the factors of the polynomial equal zero.
For the inequality \( t^4 - 4t^2 \geq 0 \), you factor it into \( t^2(t - 2)(t + 2) \), leading to potential critical points at \( t = 0, 2, -2 \).
Given the interval \([1, 6]\), only \( t = 2 \) is relevant. By considering changes around this critical point, you determine if the original inequality is satisfied on different intervals.
Understanding critical points helps in breaking the problem into smaller sections, allowing a precise determination where the polynomial meets the required conditions.

Factorization

Factorization is transforming a complex polynomial into products of simpler expressions, facilitating analysis of the expression. For \( t^4 - 4t^2 \), factor it into \( t^2(t^2 - 4) \), which further reduces to \( t^2(t - 2)(t + 2) \).
This method is essential in inequality solving as it reveals critical points, simplifying tests of intervals.
Factorizing helps identify zeroes and enables easier testing of inequality conditions since it simplifies computational complexity when substituting test points into the function.
This simplification is crucial when dealing with polynomial expressions in inequalities, leading to a more manageable approach in solving and understanding the problem.