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 the answer to two decimal places.)
Why did the triangle go to the doctor? It had a case of "hypotenuse-itis"!
Let's calculate the error in computing the length of the hypotenuse of the right triangle. We'll start by finding the derivative of the hypotenuse length with respect to the angle.
Given that one side of the right triangle is 45 cm long, let's call it the adjacent side (a), and the angle opposite that side is measured as 30°. The length of the hypotenuse (h) can be calculated using the formula:
h = a / cos(angle)
Differentiating both sides, we have:
dh = (da / cos(angle)) - a * sin(angle) * d(angle) / (cos(angle))^2
Since we know the length of the adjacent side, da = 0. Also, substituting the given values, we have a = 45 cm and angle = 30°.
dh = 0 - (45 * sin(30°) * (1°)) / (cos(30°))^2
Using a calculator, sin(30°) = 0.5 and cos(30°) = (√3) / 2.
dh = -(45 * 0.5 * (1°)) / ((√3 / 2))^2
dh = -(45 * 0.5 * (1°)) / (3 / 4)
dh = -(45 * 0.5 * 4) / 3
dh = -120 / 3
dh = -40 cm
Therefore, the estimated error in computing the length of the hypotenuse is -40 cm. However, since lengths cannot be negative, the error is rounded to two decimal places: 0.00 cm.
Oh no, looks like we made a mistake! The estimated error should be positive since the actual error would be positive when the angle is higher. So, the correct answer is the absolute value of the estimated error:
The estimated error in computing the length of the hypotenuse is 40.00 cm.
Always remember, though, errors can sometimes sneak up on you like a triangle trying to lecture you about math!
To estimate the error in computing the length of the hypotenuse, we can use differentials.
Let's denote the length of the hypotenuse as H. We have the given side length of the triangle as 45 cm and the angle opposite that side as 30°.
Now, we need to find the relationship between H and the given parameters. Since we know that the sine of an angle in a right triangle is equal to the ratio of the length of the opposite side to the length of the hypotenuse, we can write:
sin(30°) = 45 / H
To find the error in H, we can take the differential of this equation:
cos(30°) d(30°) = -45 / H^2 dH
Since the possible error in the angle is 1°, we can substitute d(30°) with 1° and cos(30°) ≈ √3/2:
(√3/2) * (1°) = -45 / H^2 dH
Simplifying the equation:
dH = (-2H^2 / (45 * √3)) * (√3/2) * (1°)
Now, we can substitute the known values to find the estimate for the error in computing the length of the hypotenuse:
dH = (-2 * (45^2) / (45 * √3)) * (√3/2) * (1°)
= - 45
Therefore, the estimated error in computing the length of the hypotenuse is -45 cm.
Note: The negative sign indicates that the estimated length of the hypotenuse is less than the actual length. However, for the purposes of estimating the error, we ignore the sign and consider the absolute value.
To estimate the error in computing the length of the hypotenuse, we can use differentials.
Let's denote the length of the hypotenuse as c and the angle as θ.
Using the trigonometric ratio, we can determine the relationship between the length of the side, the angle, and the length of the hypotenuse:
sin θ = opposite / hypotenuse
Substituting the given values:
sin 30° = 45 / c
To find the differential equation, we can take the derivative with respect to θ:
d(sin θ) = d(45 / c)
Next, let's find the derivative of sin θ:
cos θ dθ = (-45 / c^2) dc
Now, let's rearrange the equation to solve for the differential, dc:
dc = (-45c^2 / cos θ) dθ
We are given that the angle θ has a possible error of 1°. To estimate the error in computing the length of the hypotenuse, we need to calculate dc.
Since we are given the value of cos θ, we can substitute it into the equation:
cos 30° ≈ 0.866
dc = (-45c^2 / 0.866) dθ
With the values given, we have:
dc = -51.96c^2 dθ
Now, we can calculate the error in computing the length of the hypotenuse by substituting the values. Let's assume c is approximately 50 cm:
dc = -51.96(50)^2 dθ
dc = -129900 dθ
As the error in the angle θ is given as 1°, we substitute dθ = 1 into the equation:
dc = -129900
Therefore, the estimated error in computing the length of the hypotenuse is approximately -129900 cm.
Note: The negative sign indicates that the error is a decrease in length. In practice, we should take the absolute value of the answer, as errors are typically given as positive values.
hypotenuse*sintheta=45
dh sintheta+ h*costheta dtheta=0
dh= -h*ctnTheta dtheta
change dTheta to radians (2pi/180)
dh=-45cm*ctn30* 2PI/180
check my thinking