Problem 19 - Precalculus

Question

In the text we said that the solutions of the quadratic equation \(0=6+\left(88 \sin 35^{\circ}\right) t-16 t^{2}\) are \(t \approx 3.27\) and \(t \approx-0.11 .\) Use the quadratic formula and your calculator to verify these results.

Step-by-Step Solution

Answer

Values obtained are \( t \approx 3.27 \) and \( t \approx -0.11 \), confirming the given results with the quadratic formula.

Step 1: Identify the Coefficients

For the given quadratic equation \( 0 = 6 + (88 \sin 35^{\circ}) t - 16t^2 \), first identify the coefficients. Rewriting in the standard form \( at^2 + bt + c = 0 \), we have:\( a = -16 \), \( b = 88 \sin 35^{\circ} \), and \( c = 6 \).

Step 2: Calculate \( b \) using \( \sin 35^{\circ} \)

Calculate \( b = 88 \cdot \sin 35^{\circ} \) using a calculator. The sine value for \( 35^{\circ} \) is approximately \( 0.5736 \). Thus,\( b = 88 \times 0.5736 = 50.4768 \).

Step 3: Substitute into the Quadratic Formula

Substitute the values obtained into the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This becomes:\[ t = \frac{-50.4768 \pm \sqrt{50.4768^2 - 4(-16)(6)}}{2(-16)} \].

Step 4: Calculate the Discriminant

Calculate the discriminant \( D = b^2 - 4ac \):\[ D = 50.4768^2 - 4(-16)(6) = 2547.9439 + 384 = 2931.9439 \].

Step 5: Simplify the Expression

Find the values of \( t \) by simplifying the expression:\[ t = \frac{-50.4768 \pm \sqrt{2931.9439}}{-32} \].

Step 6: Solve for \( t \) Values

Using a calculator, find:\( \sqrt{2931.9439} \approx 54.1569 \).Solve for \( t \):\( t = \frac{-50.4768 + 54.1569}{-32} \approx -0.1153 \),\( t = \frac{-50.4768 - 54.1569}{-32} \approx 3.2719 \).

Step 7: Round the Case Values

Round the values to two decimal places. Hence,\( t \approx -0.11 \) and \( t \approx 3.27 \).

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool in algebra for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\).
This formula provides an efficient way of finding the roots of these equations and is applicable when factoring is difficult or impossible. To recall, the quadratic formula is given by:

• \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this formula:

• \(a\), \(b\), and \(c\) are coefficients from the equation.
• The symbol \(\pm\) indicates that there will be two possible solutions, one with addition and one with subtraction.

This process requires careful calculation, especially when handling the discriminant part under the square root. By substituting the coefficients correctly into the quadratic formula, you can solve for the variable \(t\).

Trigonometric Functions

In problems like the given exercise, trigonometric functions, such as sine, often appear due to the physical context (e.g., motion problems).
The sine function \(\sin\theta\) relates the angle \(\theta\) to the ratios of sides in a right triangle or to angles within a circle.
In this particular problem, \(b\) involved \(88 \sin 35^\circ\). Here are some key concepts related to the sine function:

• The sine value \(\sin 35^\circ\) can be found using a calculator, which provides an approximate value of \(0.5736\).
• It is crucial to always ensure your calculator is in the correct mode (degrees or radians).
• The sine function is a fundamental trigonometric function useful in many applications beyond just triangles, including waves and oscillations.

Understanding how trigonometric functions factor into equations helps grasp their broader application.

Discriminant

The discriminant is a specific part of the quadratic formula's calculation. Found under the square root in \(\sqrt{b^2 - 4ac}\), the discriminant, \(D = b^2 - 4ac\), tells us about the nature of the roots. Here are some helpful points:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is exactly one real root, meaning the solutions are the same (a perfect square).
• If \(D < 0\), there are no real roots; instead, there are two complex conjugate roots.

For our problem, we calculated \(D = 2931.9439\), which is positive, indicating two real solutions. Understanding the discriminant helps predict the nature of solutions even before fully solving the equation.

Problem-Solving Steps

Effective problem-solving in quadratic equations involves a sequence of logical and calculative steps.
Here are the simplified steps we followed in our example:1. **Identify the Coefficients:** Recognize \(a\), \(b\), and \(c\) from the given equation.
2. **Calculate \(b\) if necessary:** Some coefficients might require supplementary calculations, such as multiplying by trigonometric values.3. **Use the Quadratic Formula:** Substitute the values into the formula and solve the expression.4. **Calculate the Discriminant:** Check \(b^2 - 4ac\) to understand the nature of the solutions.5. **Solve and Simplify:** Calculate the roots using arithmetic operations and a calculator.6. **Round or Choose Appropriate Values:** Depending on the context, rounding might be necessary.These steps ensure an organized approach and reduce errors. Practicing these steps across different quadratic problems will enhance your problem-solving skills and speed.