The measurements of the base and altitude of a triangle are found to be 36 and 50 centimeters, respectively. The possible error in each measurement is 0.25 centimeter. Use differentials to approximate the possible propagated error in computing the area of the triangle.
Well, let's break it down. The formula for the area of a triangle is A = (1/2) * base * altitude.
First, let's find the differential for the base (db) and the differential for the altitude (da). Since the possible error in each measurement is 0.25 centimeter, we can say that db = 0.25 cm and da = 0.25 cm.
Next, let's find the differential for the area (dA) using the formula for differentials:
dA = (1/2) * base * da + (1/2) * altitude * db.
Substituting the given values and differentials:
dA = (1/2) * 36 cm * 0.25 cm + (1/2) * 50 cm * 0.25 cm.
Calculating that out gives us:
dA = 4.5 cm² + 6.25 cm².
So, the possible propagated error in computing the area of the triangle is approximately 10.75 cm².
Remember, math can be tri-angle, but it's all about staying sharp!
To use differentials to approximate the possible propagated error in computing the area of the triangle, we can use the formula for the area of a triangle:
Area = 1/2 * base * altitude
Let's call the base measurement B, the altitude measurement A, and the possible error in each measurement d(B) and d(A), respectively.
Given:
Base measurement (B) = 36 centimeters
Altitude measurement (A) = 50 centimeters
Possible error in base measurement (d(B)) = 0.25 centimeters
Possible error in altitude measurement (d(A)) = 0.25 centimeters
To find the propagated error in the area of the triangle, we need to differentiate the area formula with respect to both B and A, and then multiply by the respective errors:
d(Area) = (d(1/2 * B * A)) = (d(B) * 1/2 * A) + (d(A) * 1/2 * B)
Substituting the given values into the formula:
d(Area) = (0.25 * 1/2 * 50) + (0.25 * 1/2 * 36)
Simplifying the equation:
d(Area) = 6.25 + 4.5
d(Area) = 10.75 square centimeters
Therefore, the possible propagated error in computing the area of the triangle is approximately 10.75 square centimeters.
To find the propagated error in computing the area of the triangle, we can use differentials. Here, we are given the measurements of the base and altitude of the triangle, as well as the possible error in each measurement.
Let's denote the base as 'b' (36 cm) and the altitude as 'h' (50 cm). The area of a triangle, denoted by 'A', is given by the formula: A = (1/2) * b * h.
To approximate the propagated error, we can use differentials. The differential of the area, dA, can be expressed as:
dA ≈ (∂A/∂b) * db + (∂A/∂h) * dh
where (∂A/∂b) denotes the partial derivative of A with respect to b, and (∂A/∂h) denotes the partial derivative of A with respect to h.
First, let's find (∂A/∂b):
∂A/∂b = (∂(1/2) * b * h)/∂b
= 1/2 * h
Next, let's find (∂A/∂h):
∂A/∂h = (∂(1/2) * b * h)/∂h
= 1/2 * b
Now, substitute the given values into the formulas:
∂A/∂b = 1/2 * 50
= 25
∂A/∂h = 1/2 * 36
= 18
Since we are given that the possible error in each measurement is 0.25 cm, we can substitute these values into the differential equation:
dA ≈ 25 * 0.25 + 18 * 0.25
≈ 6.25 + 4.5
≈ 10.75
Therefore, the possible propagated error in computing the area of the triangle is approximately 10.75 square centimeters.
A=.5bh
dA=.5b*dh +h*.5db
Error = dA= .5(36)(.25)+.5(50)(.25)=10.75
Note: with given measurements, A = .5*36*50=900
If you are off by .25, A=.5*36.25*50.25=910.78125. Therefore you'd be off by 10.78125. Does this help to see how your estimate with differentials makes sense?