Using the formula for the circumference of a circle, C = 2πr, where C is the circumference and r is the radius, we can find the radius:

15π = 2πr

Dividing both sides by 2π, we get:

r = 7.5

Now we can find the area of the circle using the formula A = πr²:

A = π(7.5)²
A = 56.25π

The area of the circle is 56.25π.

Let x be the width of the rectangle. The length will then be 2x - 2.

The area of the rectangle can be found using the formula length * width. We are given that the area is 70, so:

x * (2x - 2) = 70

Expanding and simplifying the equation, we get:

2x^2 - 2x - 70 = 0

This quadratic equation can be factored as (x - 7)(2x + 10) = 0. Thus, x = 7 and x = -5. Since the width cannot be negative, x = 7.

Now, we can find the length:

2x - 2 = 2 * 7 - 2 = 14 - 2 = 12.

The length of the rectangle is 12.

To find the value of y:

Divide both sides of the equation by 7:
y = 63 / 7
y = 9

So, y equals 9.

To find the mean age of data set F, we need to find the total ages in data sets D and E, and then divide by the total number of people.

Total ages in data set D = 60 * 25 = 1500 years
Total ages in data set E = 40 * 37 = 1480 years

Total ages in data set F = 1500 years + 1480 years = 2980 years

Mean age in data set F = total ages / total people = 2980 years / 100 people = 29.8 years.

The mean age of data set F is 29.8 years.

The sale price is 70% less than the regular price, so the sale price is 30% of the regular price:

Sale price = 0.30 * $15.90 = $4.77

Let's call the store's cost for the shoes C. The sale price is 40% greater than the cost, so we have:

Sale price = C + 0.40 * C
$4.77 = C + 0.40C
$4.77 = 1.40C

Dividing both sides by 1.40:

C ≈ $4.77 / 1.40
C ≈ $3.41

The store's cost for the shoes was approximately $3.41.