use differentials to determine by approximately how many centimeters does the diagonal of a square table increase if its area is increased from 50 square centimeters to 54.45 square centimeters?
Area= s^2
Diagonal= sqrt(2s^2)
so, D= sqrt(2A)
dD=A^(-1/2) dA
dD= 50^(-1/2)* 4.45
dD=0.62932...
BUT IT SAYS THIS ISNT THE CORRECT ANSWER
if the diagonal is d and the area is a, then we have
d^2 = 2s^2
a = s^2 = d^2/2
now forget all that square root stuff, and use implicit differentiation:
da = d dd
we have
a = 50, so d = 10
da = 4.45
4.45 = 10 dd
dd = 0.45
You made a mistake in the line
dD=A^(-1/2) dA
It should be
dD=(2A)^(-1/2) dA
To find the approximate increase in the diagonal of the square table, we can use differentials. The differential of the diagonal, dD, can be calculated using the differential of the area, dA.
First, we know that the area of a square table is given by A = s^2, where s is the length of one side of the square.
Taking the derivative of both sides with respect to the area, we get:
dA = 2s * ds
Next, we can express the diagonal D of the square table in terms of the side length s:
D = sqrt(2s^2)
Again, taking the derivative of both sides, we get:
dD = (1/2) * (2 / sqrt(2s^2)) * (2s * ds)
= s ds / sqrt(2s^2)
= (s / sqrt(2s^2)) * ds
= (1 / sqrt(2)) * ds
Now we have expressed the increase in diagonal length dD in terms of the change in area dA. To find the approximate increase in the diagonal length, substitute the given values:
dD = (1 / sqrt(2)) * dA
= (1 / sqrt(2)) * 4.45
≈ 3.14 cm (rounded to two decimal places)
Therefore, the diagonal of the square table increases by approximately 3.14 centimeters when the area increases from 50 square centimeters to 54.45 square centimeters.
It's worth noting that your initial calculation of dD as 0.62932 was incorrect because you didn't account for the change in the area, dA. By multiplying 50^(-1/2) by 4.45, you only considered the change in area but not its influence on the increase in the diagonal length.