Area = side × side

In this case, we know the area of the square is 400 square miles. Let’s denote the length of each side as s. Therefore, we can set up the equation:

s × s = 400

This simplifies to:

s² = 400

To find the value of s, we need to take the square root of both sides:

s = √400

Calculating the square root of 400 gives us:

s = 20

So, the length of each side of the square is 20 miles. This means if you were to measure each side of the square, it would be 20 miles long.

When we do this, we have:

√(4(0)² + (0)) = √(0) = 0

So, the limit is:

0

This limit tells us about the behavior of the function as h approaches 0. Essentially, as h gets very close to 0, the entire expression inside the square root also approaches 0, leading to the final result of the limit being 0.

In parallelogram PQRS, we can identify the lengths of sides SR and QR. Given that SR measures 42 units (as it is opposite to PQ), and QR measures 10 units (as it is opposite to PS), we can substitute these values into the formula.

So, the calculation for the perimeter is:

• Length (SR) = 42
• Width (QR) = 10

Now plug in the values:

Perimeter = 2 × (42 + 10) = 2 × 52 = 104 units.

Therefore, the perimeter of parallelogram PQRS is 104 units.

1. First, arrange the equations:

Equation 1: 4x + 2y = 3
Equation 2: 3x + y = 6

2. Next, we want to eliminate one variable. We can do this by multiplying Equation 2 by 2 to match the coefficient of y in Equation 1. This gives us:

Equation 2: 2(3x + y) = 2(6)
=> 6x + 2y = 12

3. Now we have:

Equation 1: 4x + 2y = 3
Equation 3: 6x + 2y = 12

4. We can subtract Equation 1 from Equation 3 to eliminate y:

(6x + 2y) - (4x + 2y) = 12 - 3
=> 2x = 9

5. Solve for x:

x = 9/2

6. Now that we have the value of x, we can substitute it back into one of the original equations to find y. We’ll use Equation 2:

3(9/2) + y = 6
=> 27/2 + y = 6
=> y = 6 - 27/2
=> y = 12/2 - 27/2
=> y = -15/2

7. Therefore, the solution to the system of equations is:

(x, y) = (9/2, -15/2)

This means that when you plot the lines represented by these equations, they will intersect at the point (4.5, -7.5).

We can express this mathematically as:

c² = a² + b²

Substituting in the values for a and b:

c² = 8² + 15²

Calculating the squares:

c² = 64 + 225

c² = 289

To find c, we take the square root of both sides:

c = √289

c = 17

Therefore, the length of the hypotenuse is 17 units.

If we divide each term by its predecessor, we get:

• 12 ÷ 3 = 4
• 48 ÷ 12 = 4
• 192 ÷ 48 = 4

This shows that each term is obtained by multiplying the previous term by 4. Thus, it is clear that the sequence is generated by a common ratio of 4. The explicit rule for the nth term can therefore be expressed as:

a_n = 3 × 4^(n-1)

Where:

• a_n is the nth term of the sequence.
• n is the position of the term in the sequence (starting from n = 1).

Let’s confirm this rule by calculating the first few terms:

• For n = 1: a₁ = 3 × 4^(1-1) = 3 × 1 = 3
• For n = 2: a₂ = 3 × 4^(2-1) = 3 × 4 = 12
• For n = 3: a₃ = 3 × 4^(3-1) = 3 × 16 = 48
• For n = 4: a₄ = 3 × 4^(4-1) = 3 × 64 = 192

As expected, our calculated terms match the original sequence. Thus, the explicit rule for the nth term of the sequence is confirmed: a_n = 3 × 4^(n-1).

Follow these steps:

1. Identify the function: Let’s denote the function as f(x).
2. Differentiate the function: Calculate f'(x), the derivative of f with respect to x. This can be done using basic differentiation rules.
3. Evaluate the derivative at the given point: Substitute the x-coordinate of the indicated point into the derivative f'(x) to find the slope at that specific point.

For example, if we have a function f(x) = x² and we want to find the slope at the point (2, f(2)), we first find the derivative f'(x) = 2x. Then, substituting x = 2 gives us f'(2) = 2(2) = 4. Thus, the slope of the curve at the point (2,4) is 4.

Using this method will allow you to find the slope of any curve at any point you are interested in.

P(A ∩ B) = P(A) × P(B)

Given that:

• P(A) = 0.4
• P(B) = 0.6

Now, substituting the values into the formula:

P(A ∩ B) = 0.4 × 0.6 = 0.24

Therefore, the probability of both events A and B occurring together is P(A ∩ B) = 0.24.

(x + 3)/9 × 6/(x + 12)

Now, we’ll multiply the fractions:

=(x + 3) × 6 / (9 × (x + 12))

This can now be simplified further. We can reduce the coefficients:

6/9 = 2/3

So our expression becomes:

=(x + 3) × 2 / (3 × (x + 12))

This gives us:

=(2(x + 3)) / (3(x + 12))

Therefore, the simplified form of the expression is:

=(2x + 6) / (3x + 36)

(x – h)² + (y – k)² = r²

In this equation, (h, k) represents the coordinates of the center of the circle, and r is the radius.

For a circle with center at (10, 6) and radius 6, we can substitute these values into the equation:

• h = 10
• k = 6
• r = 6

Now, substituting into the standard equation:

(x – 10)² + (y – 6)² = 6²

This simplifies to:

(x – 10)² + (y – 6)² = 36

So, the standard equation for the circle centered at (10, 6) with a radius of 6 is:

(x – 10)² + (y – 6)² = 36