Video: Solving Trigonometric Equations Modeling Real-Life Situations by Direct Substitution

The height, ℎ, of a piston can be modeled by the equation ℎ = 2cos 𝑥 + 6, where 𝑥 represents the crank angle and ℎ is measured in inches. Find, to 2 decimal places, the height of the piston when the crank angle is 55°.

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Video Transcript

The height, ℎ, of a piston can be modeled by the equation ℎ equals two cos 𝑥 plus six, where 𝑥 represents the crank angle and ℎ is measured in inches. Find, to two decimal places, the height of the piston when the crank angle is 55 degrees.

We’ll start with the given equation: ℎ equals two cos 𝑥 plus six. ℎ equals the height of the piston in inches, and 𝑥 equals the crank angle. We’re given that the crank angle is 55 degrees. That’s our 𝑥 value. So we plug in 55 degrees for 𝑥. And the ℎ value is what we’re trying to find. We want to know what the height of the piston would be to the nearest two decimal places.

If we plug in two times cos of 55 degrees in the calculator, we get a decimal value that does not terminate, 1.14715 continuing. We can add the whole number six to this value. One plus six equals seven. The new value is 7.14715 continuing.

At our last step, we need to round to two decimal places. Rounding to the nearest hundredths means we look to the right, to the thousandths place. There’s a seven. And we need to round up. Our four becomes a five, and everything to the left of the hundredths place stays the same. This formula is measured in inches, so the height of our piston must be 7 and 15 hundredths inches.

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