Explanation

To solve this problem, calculate the total hours worked by the men:

12 men × 8 hours/day × 10 days = 960 hours

Now, find the number of men needed to complete the work in 5 days, working 12 hours a day:

Let x be the number of men.

x men × 12 hours/day × 5 days = 960 hours

x × 60 = 960

x = 16

Explanation

When throwing a die, there are 6 possible outcomes (1, 2, 3, 4, 5, 6).

When drawing a letter at random from the English alphabet, there are 26 possible outcomes (A, B, C, ..., Z).

Since the two events are independent, the total number of points in the sample space is the product of the two:

6 (die outcomes) × 26 (letter outcomes) = 156

Explanation

An order on a set "S" is typically denoted by the symbol "≤" (less than or equal to).

It representing a relation that establishes a sense of ordering between elements within the set. 

Explanation

To find the slope of the tangent to the curve y³x + y²x² = 6 at (2, 1), we'll use implicit differentiation:

Given: y³x + y²x² = 6

Differentiating both sides with respect to x:

y³ + x_3y²(dy/dx) + y²_2x + x²*2y(dy/dx) = 0

Rearranging terms to isolate dy/dx:

(3xy² + 2x²y)(dy/dx) = -y³ - 2xy²

dy/dx = (-y³ - 2xy²) / (3xy² + 2x²y)

Now, evaluating dy/dx at (2, 1):

dy/dx = (-(1)³ - 2_2_(1)²) / (3_2_(1)² + 2*(2)²*1)

= (-1 - 4) / (6 + 8)

= -5 / 14

Explanation

Explanation

The derivative of 2^(-x) is -2^(-x) * ln(2), which can be written as -2⁻ˣ·log2 (considering log as ln or log base e implicitly in the derivative formula).

Explanation

Explanation

Explanation

To estimate the area under the graph of f(x) = x^2 in the interval [0,1] using a lower sum with two rectangles of equal width:

Width of each rectangle = (1 - 0) / 2 = 0.5

Rectangle 1: Height = f(0) = 0^2 = 0, Area = 0 * 0.5 = 0

Rectangle 2: Height = f(0.5) = 0.5^2 = 0.25, Area = 0.25 * 0.5 = 0.125

Total estimated area = 0 + 0.125 = 0.125

Explanation