One side of a right triangle is known to be 45 cm long and the opposite angle is measured as 30°, with a possible error of ±1°.

(a) Use differentials to estimate the error in computing the length of the hypotenuse. (Round your answer to two decimal places.)
± cm

-2.72

To estimate the error in computing the length of the hypotenuse, we can use differentials. The general formula for differentials is:

dY ≈ (∂Y/∂X) * dX

Where dY represents the change in the dependent variable Y, ∂Y/∂X represents the partial derivative of Y with respect to X, and dX represents the change in the independent variable X.

In this case, let's consider the hypotenuse (H) as our dependent variable and the angle (θ) as our independent variable. We want to estimate the error in H caused by a change in θ.

The length of the hypotenuse in a right triangle can be found using the formula:

H = (side length) / sin(θ)

Taking the derivatives of both sides with respect to θ, we have:

dH = (d(side length) / sin(θ)) - ((side length) * cos(θ) * (dθ / sin^2(θ)))

Since we are given a possible error of ±1° in θ, we can consider dθ as ±1°.

Plugging in the known values, we have:

dH = (0 / sin(30°)) - (45 * cos(30°) * (±1° / sin^2(30°)))

Using a calculator, we can find sin(30°) = 0.5 and cos(30°) = √3/2.

Calculating the expression further, we get:

dH = 0 - (45 * (√3/2) * (±1° / 0.5^2))

Simplifying, we have:

dH = -45 * (√3/2) * (±1° / 0.25)
dH = -45 * (√3/2) * (±4°)

Using the calculator once more, we get (√3/2) ≈ 0.866.

dH ≈ -45 * 0.866 * (±4°)

Now, let's calculate the upper bound and lower bound separately:

Upper bound:
dH = -45 * 0.866 * 4
dH ≈ -155.7 cm

Lower bound:
dH = -45 * 0.866 * (-4)
dH ≈ 155.7 cm

Therefore, the error in computing the length of the hypotenuse is approximately ±155.7 cm.

To estimate the error in computing the length of the hypotenuse, we can use differentials.

First, let's denote the length of the hypotenuse as H and the angle opposite the 45 cm side as θ. We want to find the error in H when there is a ±1° error in θ.

We can use the sine function to relate the angle and the sides of the right triangle:

sin(θ) = (opposite side) / (hypotenuse)
sin(θ) = 45 / H

Now, let's differentiate both sides with respect to θ:

cos(θ) * dθ = (-45 / H^2) * dH

Since the error in θ is ±1°, we have dθ = ±1° = ±π/180 radians. Let's use the positive value here to calculate the maximum error:

cos(θ) * (π/180) = (-45 / H^2) * dH

Next, we need to express cos(θ) in terms of sin(θ). Using the Pythagorean identity:

cos(θ) = sqrt(1 - sin^2(θ))
cos(θ) = sqrt(1 - (45 / H)^2)

Now, we substitute this into the previous equation:

sqrt(1 - (45 / H)^2) * (π/180) = (-45 / H^2) * dH

Finally, we can solve for dH:

dH = -((π/180) / sqrt(1 - (45 / H)^2)) * (45 / H^2)

Let's plug in the given values: θ = 30° and H = 45 cm:

dH = -((π/180) / sqrt(1 - (45 / 45)^2)) * (45 / 45^2)
dH ≈ -0.02376 cm

Rounding to two decimal places, the estimated error in computing the length of the hypotenuse is approximately ±0.02 cm.