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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. (a) Use differentials to approximate the possible propagated error in computing the area of the triangle. (b) Approximate the percent error in computing the area of the triangle. Step 1 of 3 A Consider that the measurement base and the altitude of a triangle of a triangle is 36 and 50 centimeters, respectively. Also, the possible error in each measurement is 0.25 centimeter. Comment Step 2 of 3 A (a) Objective is to approximate the propagated error in computing the area of the triangle with the help of differential. For this note that the formulae for the area of triangle is: A= (bn) Here, b is the base of triangle and h is the height of triangle. Thus, b= 36, h = 50 And, db = dh = +0.25 To approximate the propagated error differentiate the area and get dA, dA= 2 1 = 56(dh) +=n(db) - }(50)x(+0.25)+} (36)*(+0.25) = +10.75 Thus area has propagate error of about 10.75 cm? Comment Step 3 of 3 A (b) The percent error can be calculated as follows: +b(dh) +hdb -x100 dA x 100 = 2 A 1 bh 2 21.50 -x100 1800 = 1.194 Hence, the required percent error is 1.194%

Sagot :

The error in computing the area of triangle is 10.75 and the percentage error in computing the area of triangle is 1.194%.

Given that, base of the triangle is 36 cm (a)

Altitude of the triangle is 50 cm (b)

Error in each measurement is 0.25 cm

We know that, area of the triangle (s) = 1/2 * base * altitude

Let us consider base as ' a ' and altitude as ' b '

So, the maximum error of ' s ' can be calculated as

⇒ ds/da * da + ds/db * db

⇒ 1/2* b * da + 1/2 *a * db

⇒ 1/2* 36 * 0.25 + 1/2* 50* 0.25

⇒ 4.5 + 6.25 = 10.75

Now, let us calculate the percentage error in computing the area of the triangle.

dA/A * 100 = [(1/2* b* da + 1/2* a * db)/ 1/2* b *h] * 100

⇒ [(1/2* 36 * 0.25 + 1/2* 50 * 0.25)/ 1/2* 36 * 50] * 100

⇒ [ 21.5/ 1800 ] * 100

⇒ 1.194%

Thus, the error in computing the area of the triangle is 10.75 and the percentage error in computing the area of the triangle is 1.194%.

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