Use differentials to determine by approximately what percentage does the area of a square table increase if its diagonal is increased from 38 centimetres to 39.235 centimetres.
I know that the surface is A = 1/2y^2 and y = x * square root of 2 but I don't know how to solve it... Can someone help?
A = 1/2 y^2
dA = y dy ≈ y ∆y = 38 * 1.235 = 46.93
check:
1/2 * 39.235^2 - 1/2 * 38^2 = 47.69
so what would the answer be?,
To solve this problem using differentials, we can relate the change in the area of the square table to the change in its diagonal length.
Let's denote the side length of the square table as x. From the given information, we know that the diagonal length of the square table is given by d = 38 cm and d + dx = 39.235 cm.
The area of the square table can be calculated as A = x^2.
Now, let's differentiate both sides of the equation with respect to x:
dA/dx = 2x (Differentiating x^2)
From the relation y = x√2, we can substitute x with y/√2 in the equation above:
dA/dx = 2(y/√2) = √2y
Next, we want to determine the percentage change in the area (∆A) when the diagonal length (∆d) changes from 38 cm to 39.235 cm.
We can approximate this using differentials:
∆A ≈ dA = √2y × ∆d (where ∆d = 39.235 cm - 38 cm)
To find the exact value of y, we can use the relationship y = x√2 where x = side length of the square table. Since the diagonal is d = 38 cm, we can use the Pythagorean theorem:
d^2 = x^2 + x^2
38^2 = 2x^2
x^2 = 38^2 / 2
x = 19√2
Substituting this value of x into y = x√2:
y = 19√2 × √2
y = 38
Therefore, y = 38 cm.
Now we can calculate ∆A:
∆A ≈ √2 × 38 × (∆d) = √2 × 38 × (39.235 - 38)
Simplifying the expression:
∆A ≈ 1.414 × 38 × 1.235
∆A ≈ 68.2037
To find the percent change in area, divide ∆A by the original area:
% change in area = (∆A / A) × 100
% change in area = (68.2037 / (38^2)) × 100
% change in area ≈ 44.959%
Therefore, approximately, the area of the square table increases by around 44.959% when its diagonal is increased from 38 cm to 39.235 cm.