To find the percent by which the radius must be increased, we need to first understand the relationship between the height and radius of a cylindrical jar and its volume.

The volume (V) of a cylinder is given by the formula:

V = πr^2h

Where:
- V is the volume of the cylinder
- π is a constant, approximately equal to 3.14
- r is the radius of the cylinder
- h is the height of the cylinder

Given that the height of the cylindrical jar is decreased by 36%, let's assume the original height is h and the new height is 0.64h (since 100% - 36% = 64%).

We want to find the percent by which the radius (r) must be increased so that there is no change in the volume.

Let's calculate the original volume (V1) and the new volume (V2) using the given information:

Original Volume (V1) = πr^2h
New Volume (V2) = π(r + x)^2(0.64h)

Where:
- x is the increase in radius (which we need to find)
- h is the original height

Since there is no change in volume, we can equate V1 and V2:

πr^2h = π(r + x)^2(0.64h)

Let's cancel out the common terms:

r^2 = (r + x)^2(0.64)

Expanding the equation:

r^2 = (r^2 + 2rx + x^2)(0.64)

Now, let's simplify the equation:

r^2 = 0.64r^2 + 1.28rx + 0.64x^2

Subtracting r^2 from both sides:

0 = 0.64r^2 + 1.28rx + 0.64x^2 - r^2

0 = 0.64r^2 - r^2 + 1.28rx + 0.64x^2

0 = -0.36r^2 + 1.28rx + 0.64x^2

To make calculations easier, let's divide the entire equation by 0.64:

0 = -0.5625r^2 + 2rx + x^2

Now, let's simplify the equation further:

0 = -0.5625(r^2) + 2rx + x^2

0 = -0.5625r^2 + 2rx + x^2

This equation represents a quadratic equation in terms of x. To solve for x, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = -0.5625, b = 2r, and c = 0.

Substituting the values into the quadratic formula:

x = (-2r ± √((2r)^2 - 4(-0.5625)(0)))/(2(-0.5625))

Simplifying:

x = (-2r ± √(4r^2))/(2(-0.5625))
x = (-2r ± 2r)/(2(-0.5625))
x = -r/(-0.5625)