What are the coordinates of the centroid of a triangle with vertices at D (0,0), E(-8,

There are a total of 20,000 available numbers. To ascertain how many phone numbers are available, the number of digits in each number must be known. Here, I will assume each number has 7 digits. This leads to the format starting with either 373XXXX or 377XXXX, where each X can be any digit from 0 to 9. This results in 10 options for each X. Therefore, there are 10×10×10×10 = 10,000 distinct phone numbers starting with 373 and another 10,000 with 377, totaling 20,000 numbers.

Response:

a. 55 cars

b. 25 cars

Detailed explanation:

Let’s denote the quantity of cars with stereo systems as N(ss), those with air conditioning as N(ac), and those with sunroofs as N(sr).

We find that:

N(ss) = 30

N(ac) = 30

N(sr) = 40

N(ss and ac and sr) = 15

N(at least two) = 30

a.

To calculate how many cars possess at least one feature (N(at least one) or N(ss or ac or sr)), we apply:

N(ss or ac or sr) = N(ss) + N(ac) + N(sr) - N(ss and ac) - N(ss and sr) - N(ac and sr) + N(ss and ac and sr)

N(ss or ac or sr) = 30 + 30 + 40 - (N(at least two) + 2*N(ss and ac and sr)) + 15

Substituting, we find N(ss or ac or sr) = 30 + 30 + 40 - (30 + 2*15) + 15 = 55

b.

For those cars that have exactly one feature, we have:

N(only one) = N(at least one) - N(at least two)

N(only one) = 55 - 30 = 25

Approximately 2, I believe... I calculated it to be around 2.10. 

Answer:

Part 1) The equation is

Part 2) When x=40 m, the area of the schoolyard is A=2,800 m^2

Part 3) The valid domain consists of all real numbers exceeding zero and below 75 meters

Step-by-step explanation:

Part 1) Formulate an expression for A(x)

Let

x -----> the length of the rectangular school yard

y ---> the width of the rectangular school yard

It is known that

The perimeter for the fencing (taking the school wall as one side) is

thus

-----> this is equation A

The area of the rectangular school yard is

----> this is equation B

Substituting equation A into equation B yields

Change to function notation

Part 2) What is the area when x=40?

With x equal to 40 m

substitute the value into the expression from Part 1 to determine A

   

Part 3) What would be a suitable domain for A(x) in this scenario?

We understand that

A signifies the area of the rectangular school yard

x characterizes the length of the rectangular school yard

It follows that

This forms a vertical parabola opening downwards

The vertex indicates a maximum point

The x-coordinate of the vertex corresponds to the length that maximizes the area

The y-coordinate of the vertex denotes the maximum area

The vertex corresponds to (37.5, 2812.5)

Refer to the accompanying figure

Consequently,

The peak area achieved is 2,812.5 m^2

The x-intercepts are located at x=0 m and x=75 m

The domain for A is the range -----> (0, 75)

All real numbers greater than zero and less than 75 meters

Honestly, I find Mrs. Garcia's method easier to perform mentally. It hinges on how familiar you are with your multiples of 5. (5*15 = 75 is a multiplication I often use)

Melissa's approach involves calculating 5*20 = 100 and 5*9 = 45, then combines the 3-digit result 100 with the 2-digit result 45, yielding 145. Adding 45 to 00 is simple and doesn’t require carrying digits, thus the arithmetic is fairly straightforward.

Mrs. Garcia's technique involves computing 5*14 = 70 and 5*15 = 75, then summing these two-digit results. Many people may not readily recall that 5*15=75, which complicates forming that product. The addition of 70 and 75 requires a carrying operation, making the math somewhat more complex. The resulting total is 145.

(The rationale behind my preference for Mrs. Garcia's method is that I can achieve the final sum by simply doubling 7 tens, followed by adding 5. The only 3-digit number to remember mentally is the ultimate total.)

_____Subtraction introduces a slight complication, yet reshaping it as $5(30 -1) = $150 - 5 = $145 is possible.
Or, you may reframe it as $5(28 +1) = $140 +5 = $145.
Dividing an even number by 2 to find the product of 5 is straightforward when you append a zero.
  5*14 = 10*7 = 70
  5*28 = 10*14 = 140.