Math Snap

STEP 1

1. The equation $6 \cos \left(\frac{\pi}{4} x\right) = 5.5$ is a trigonometric equation.
2. We are looking for solutions within the interval $[0, 6]$.
3. The cosine function is periodic, and we may have multiple solutions within the given interval.

STEP 2

1. Isolate the cosine function.
2. Solve for the angle.
3. Find all solutions for $x$ in the given interval.

STEP 3

First, divide both sides of the equation by 6 to isolate the cosine function:
$$6 \cos \left(\frac{\pi}{4} x\right) = 5.5$$ $$\cos \left(\frac{\pi}{4} x\right) = \frac{5.5}{6}$$ $$\cos \left(\frac{\pi}{4} x\right) = \frac{11}{12}$$

STEP 4

Now, solve for the angle $\frac{\pi}{4} x$ by taking the inverse cosine (arccos) of both sides:
$$\frac{\pi}{4} x = \cos^{-1} \left(\frac{11}{12}\right)$$ Calculate the principal value of $\cos^{-1} \left(\frac{11}{12}\right)$:
Let $\theta = \cos^{-1} \left(\frac{11}{12}\right)$.

STEP 5

Since the cosine function is periodic with period $2\pi$, the general solutions for the angle are:
$$\frac{\pi}{4} x = \theta + 2k\pi \quad \text{and} \quad \frac{\pi}{4} x = -\theta + 2k\pi$$ where $k$ is an integer.

STEP 6

Solve for $x$ in both cases:
1. $\frac{\pi}{4} x = \theta + 2k\pi$
$$x = \frac{4}{\pi}(\theta + 2k\pi)$$ 2. $\frac{\pi}{4} x = -\theta + 2k\pi$
$$x = \frac{4}{\pi}(-\theta + 2k\pi)$$ Calculate the specific values of $x$ for $k$ such that $x$ is in the interval $[0, 6]$.

STEP 7

Calculate $\theta = \cos^{-1} \left(\frac{11}{12}\right)$ using a calculator:
$\theta \approx 0.4510$ radians.
Substitute $\theta$ back into the equations for $x$:
1. $x = \frac{4}{\pi}(0.4510 + 2k\pi)$
2. $x = \frac{4}{\pi}(-0.4510 + 2k\pi)$
Test integer values of $k$ to find solutions in the interval $[0, 6]$.

For $k = 0$:
1. $x \approx \frac{4}{\pi}(0.4510) \approx 0.573$
2. $x \approx \frac{4}{\pi}(-0.4510) \approx -0.573$ (not in the interval)
For $k = 1$:
1. $x \approx \frac{4}{\pi}(0.4510 + 2\pi) \approx 5.146$
2. $x \approx \frac{4}{\pi}(-0.4510 + 2\pi) \approx 4.573$
The solutions within the interval $[0, 6]$ are:
$$x = 0.573, 4.573, 5.146$$ The solutions for $x$ in the interval $[0, 6]$ are:
$$\boxed{0.573, 4.573, 5.146}$$