use differentials to determine by approximately how many square centimeters does the area of a square table increase if its diagonal is increased from 4 centimeters to 4.092 centimeters
the question is
The area of table increases by approximately (??) square centimetres
I'm doing my assignment but we did not cover these stuff yet and they but them In our online assignment
well Diagonal = D = x sqrt 2
so dD = sqrt 2 dx
a = x^2
so
da = 2 x dx etc
Thank You very much Mr Damon
To determine the approximate increase in the area of a square table when its diagonal is increased, we can use differentials.
Let's denote the area of the square table as A and the diagonal as D. We are given that the initial diagonal D1 is 4 centimeters, and the new diagonal D2 is 4.092 centimeters.
The relationship between the side length of a square (s) and its diagonal (D) is given by the Pythagorean theorem: D^2 = 2s^2. Rearranging this equation, we can solve for the side length of the table: s = (D^2)/2.
Now, we need to find the relationship between the area (A) and the side length (s). Since we know that the table is square, the area is simply A = s^2.
To find the increase in the area, we can compute the differential of the area, dA. We can do this by taking the derivative of A with respect to s, and then multiplying it by the differential of s, ds. We obtain: dA = 2s * ds.
To find the approximate increase in the area, ΔA, we substitute Δs for ds, since we are interested in the change in s and not a specific value of ds. Therefore, ΔA = 2s * Δs.
Now, we can calculate the side lengths and areas for both the initial diagonal and the increased diagonal:
For D1 = 4 cm, s1 = (4^2)/2 = 8 cm, and A1 = s1^2 = 64 cm^2.
For D2 = 4.092 cm, s2 = (4.092^2)/2 = 8.357 cm, and A2 = s2^2 = 69.966 cm^2.
The approximate increase in the area, ΔA, is calculated as follows:
ΔA = A2 - A1 = 69.966 cm^2 - 64 cm^2 = 5.966 cm^2 (rounded to 3 decimal places).
Therefore, the area of the square table increases by approximately 5.966 square centimeters when its diagonal is increased from 4 centimeters to 4.092 centimeters.