The measure of each exterior angle of a regular polygon can be calculated using the formula 360/n, where n is the number of sides. For a dodecagon (12 sides), it is 360/12 = 30 degrees.

The measure of each exterior angle of a regular polygon is calculated as 360/n, where n is the number of sides. For a dodecagon (12 sides), it is 360/12 = 30 degrees.

Using the formula n(n-3)/2, for an octagon (n=8), the number of diagonals is 8(8-3)/2 = 20.

The measure of each interior angle of a regular polygon can be calculated using the formula [(n-2) * 180] / n. For an octagon (n=8), it is [(8-2) * 180] / 8 = 135 degrees.

The sum of the interior angles of an octagon (8-sided polygon) is calculated using the formula (n-2) * 180, which gives (8-2) * 180 = 1080 degrees.

In the polynomial P(x), the coefficient of the x^2 term is -4.

In the polynomial P(x), the coefficient of x^2 is -4.

The roots of the polynomial can be found by factoring it as (x - 2)(x - 3) = 0, giving roots 2 and 3.

The roots of the polynomial can be found by factoring it as (x - 2)(x - 3) = 0, giving roots 2 and 3.

The roots of the polynomial can be found by factoring it as (x - 2)(x - 3) = 0, giving roots 2 and 3.

By applying the Rational Root Theorem and synthetic division, we can find that p(x) has three distinct real roots.

Expected affected individuals = Prevalence × Population = 0.03 × 1000 = 30.

Population after 2 years = 1000 * (1 + 0.10)^2 = 1000 * 1.21 = 1210.

After 1 year, the population will be 1000 * 1.15 = 1150. After 2 years, it will be 1150 * 1.15 = 1322.5, rounded to 1325.

Let the initial population be 100. After a 15% increase, it becomes 115. After a 10% decrease, it becomes 115 - 11.5 = 103.5. The net change is (103.5 - 100) / 100 * 100% = 3.5%.

Population after 1 year = 1,000 * 1.05 = 1,050. After 2 years = 1,050 * 1.05 = 1,102.5.

Increasing the length of the wire increases the potential gradient, which can improve the accuracy of the measurement by allowing for finer adjustments.

For accurate comparison of two emfs using a potentiometer, both circuits must have the same potential gradient to ensure that the readings are directly comparable.

The potential gradient is calculated as the potential difference divided by the length of the wire: 12 V / 6 m = 2 V/m.

The EMF of the cell can be calculated as EMF = potential gradient × length = 1.5 V/m × 3 m = 4.5 V.

To measure the potential difference across a resistor, the potentiometer must be connected in parallel with the resistor.

The potential gradient is calculated as the potential difference divided by the length of the wire: 12 V / 6 m = 2 V/m.

The potential gradient is calculated as V/L = 12V/4m = 3 V/m.

Using Ohm's law, V = IR = 0.5 A * 10 ohms = 5 V.

In 15 minutes, the printer can print 20 pages/minute * 15 minutes = 300 pages.

Loss = Cost Price - Selling Price = 150 - 120 = 30. Loss Percentage = (Loss/Cost Price) * 100 = (30/150) * 100 = 20%.

Loss = Cost Price - Selling Price = $150 - $120 = $30. Percentage loss = (Loss/Cost Price) * 100 = ($30/$150) * 100 = 20%.

Selling Price = $500 - (30% of $500) = $500 - $150 = $350. Since Selling Price = Cost Price, there is no profit or loss.

Discount = Marked Price - Selling Price = 500 - 450 = 50. Discount Percentage = (50/500) * 100 = 10%.

Let the cost price be x. Then, 80 = x - (20% of x) => 80 = x - 0.2x => 80 = 0.8x => x = 100.