Given a rectangular object with dimensions 2cm by 22cm, we want to increase the length of the smaller dimension and decrease the length of the larger dimension by Xcm each.

If the the diagonal has to be kept less than (sqrt360)cm in length, the range of X can be expressed as a<X<b. What are the values of a and b?

(2+x)^2 + (22-x)^2 < 360

4 < x < 16

Thanks for your help in all the questions

To determine the range of X, we need to find the maximum and minimum values of X such that the diagonal length remains less than (sqrt360) cm.

First, let's calculate the length of the diagonal of the original rectangular object.
Using the Pythagorean theorem, the length of the diagonal (D) can be found using the formula:
D² = (length)² + (width)²

Given that the length is 22 cm and the width is 2 cm:
D² = (22)² + (2)²
D² = 484 + 4
D² = 488
D ≈ 22.09 cm

To maintain the diagonal length less than (sqrt360) cm:
D < sqrt(360)
22.09 < sqrt(360)

Now, let's analyze the effect of increasing the smaller dimension by X cm and decreasing the larger dimension by X cm.

For the diagonal length to remain less than (sqrt360) cm, we need to find the maximum and minimum value of X that satisfy the condition.

1. Maximum value of X:
To maximize X, we would reduce the larger dimension by X as much as possible. To keep the diagonal less than (sqrt360) cm, we have the equation:
(sqrt360) > sqrt((length+X)² + (width-X)²)

Simplifying the equation:
360 > (length+X)² + (width-X)²

Using the initial dimensions (length = 22 cm and width = 2 cm):
360 > (22+X)² + (2-X)²

Expanding and simplifying:
360 > X² + 44X + 484 + X² - 4X + 4

Combining like terms:
0 > 2X² + 36X + 120

To solve the quadratic inequality, we can find the roots of the equation by setting it equal to zero:
2X² + 36X + 120 = 0

Using the quadratic formula:
X = (-b ± sqrt(b² - 4ac)) / 2a

a = 2, b = 36, c = 120
X = (-36 ± sqrt(36² - 4(2)(120))) / 2(2)
X = (-36 ± sqrt(1296 - 960)) / 4
X = (-36 ± sqrt(336)) / 4
X ≈ (-36 ± 18.33) / 4

Taking the positive root:
X ≈ (-36 + 18.33) / 4
X ≈ -17.67 / 4
X ≈ -4.42

Since X represents a decrease in length, the maximum value of X that satisfies the condition is approximately -4.42 cm. However, we need to consider the practical context, so we'll consider the maximum value of X to be 0 cm.

2. Minimum value of X:
To minimize X, we would reduce the smaller dimension by X as much as possible. To keep the diagonal less than (sqrt360) cm, we have the equation:
(sqrt360) > sqrt((length-X)² + (width+X)²)

Simplifying the equation:
(sqrt360) > sqrt((22-X)² + (2+X)²)
360 > (22-X)² + (2+X)²

Expanding and simplifying:
360 > X² - 44X + 484 + X² + 4X + 4

Combining like terms:
0 > 2X² - 40X + 120

To solve the quadratic inequality, we can find the roots of the equation by setting it equal to zero:
2X² - 40X + 120 = 0

Using the quadratic formula:
X = (-b ± sqrt(b² - 4ac)) / 2a

a = 2, b = -40, c = 120
X = (-(-40) ± sqrt((-40)² - 4(2)(120))) / 2(2)
X = (40 ± sqrt(1600 - 960)) / 4
X = (40 ± sqrt(640)) / 4
X ≈ (40 ± 25.3) / 4

Taking the positive root:
X ≈ (40 + 25.3) / 4
X ≈ 65.3 / 4
X ≈ 16.33

Therefore, the minimum value of X that satisfies the condition is approximately 16.33 cm.

In summary, the range of X can be expressed as a < X < b, where a ≈ -4.42 cm and b ≈ 16.33 cm.