In an arithmetic series, the last term can be expressed as a + (n-1)d. Here, 48 = 12 + (10-1)d. Thus, 48 - 12 = 9d, giving d = 4.

Setting y = 0 gives 2x = 6, thus x = 3. The point of intersection is (3, 0).

Setting y = 0 gives 2x = 6, thus x = 3. The x-coordinate of the intersection point is 3.

Setting x = 0 gives 4y = 12, thus y = 3. The point of intersection is (0, 3).

Setting x = 0 gives -4y + 12 = 0, thus y = -3. The point of intersection is (0, -3).

Rearranging gives y = (3/4)x + 3. The slope of the line is 3/4, so the slope of the parallel line is -3/4.

Parallel lines have the same slope. The slope of the given line is 3/4, which is the same as the slope of 6x - 8y + 24 = 0.

Rearranging to y = -7/2x + 7 shows that the slope is -7/2.

The product of the roots gives k = (-2)(-3) = 6.

Original area = π(10)² = 314 cm²; New area = π(8)² = 201.06 cm². Change = 314 - 201.06 = 112.94 cm².

Area change = π[(r-2)² - r²] = π[-4r + 4] = 3.14 * (-4r + 4).

Area = πr²; if r is doubled, area = π(2r)² = 4πr², so it quadruples.

Circumference = 2πr; If r is halved, new circumference = πr; Factor = 1/2.

Circumference = 2πr. If radius is halved, new circumference = 2π(r/2) = πr, which is halved.

Circumference = 2πr. If radius is halved, new circumference = 2π(r/2) = πr, which is halved.

Area increases by a factor of (3r)²/r² = 9.

Area = πr². If radius is tripled, new area = π(3r)² = 9πr², which is 9 times the original area.

Max revenue occurs at x = -b/(2a) = 100/(2*2) = 25.

R'(x) = 20 - x = 0 gives x = 20. This maximizes revenue.

Using the sum and product of roots: 1 + (-3) = -2 and 1 * (-3) = -3, thus k = 3.

The discriminant must be greater than zero: 2² - 4*1*k > 0, which simplifies to k < 1.

The sum of the roots is given by -b/a = -5/1 = -5.

Using the product of roots: k = (-2)(-3) = 6.

Using the sum and product of roots: k = 1*4 = 4, and sum = 1 + 4 = 5, thus k = 7.

Using the product of roots: q = 1 * 4 = 4.

Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find k = 12.

Using the product of roots, k = 1*6 = 6.

Using the sum of roots (-3 + -4 = -7) and product of roots (-3*-4 = 12), we find p = 12.

Using the product of roots: p = (-3)(-4) = 12.

Using the sum of the roots: p = -(3 + 4) = -7.