KingHTML

Good Job EP, I got the answer of 90 (92.5-2.5) as well!

I'm pretty sure that this is correct, Nice Try!

I'm not a 100 percent sure on this one, so you may wanna check my work before.  

Let's first start with the most restrictive restriction:

- There has to be a repeating pattern of 3 digits that is going to be repeated either by the second loop or the third loop, and stay in the first loop. 

The repeating pattern has 1600 possibilities because there are 8 possibilities for the hundreds digit (2-9), and 10 possibilities for the ones and tens digit (0-10). 

8 * 10 * 10 = 800

We have to multiply by 2, because the repeating pattern can either be in the second loop of 3 digits or the 2nd one. 

Therefore 800*2 = 1600 phone numbers

Problem: x/13 = 69

First step: Multiple 13 on both sides

x/13 * 13 = 69 * 13

x = 69*13

x = 897

Therefore the missing variable (x) is 897. 

We can assign the length and width variables, l and w respectively. 

The increase in each dimension would result in:

(l*1.2) * (w*1.2) = lw * 1.44

Since we know that lw is the original area, there is a 44 percent increase. 

We don't even need to consider the fact that the dimensions are 24 and 16!

Ako, I understand your process, but the last part of arithmetic is wrong. 

We know that the first term in the sequence in numbers between is 17/3 which's ceiling is 6, and (17/3)^2, which's floor is 32. 

The sequence is as follow: 

(6, 7, 8, ...., 30, 31, 32)

So we have to find the number of terms between 6 and 32 inclusive, which is 32 - 6 + 1 and that is 27, we have to do this because subtraction only includes 1 of the values not both of them. 

Or we could've done it by subtracting 5 from every term in the sequence then you would get:

(6-5, 7-5, 8-5, ...., 30-5, 31-5, 32-5)

(1, 2, 3, ..., 25, 26, 27)

Either way you can see that there are 27 terms, so the answer is 27 integers. 

We know that working 8 hours per day for 18 days results in 144 days. 

Since we are increasing the number of typists by 16/12 or 4/3 the number of hours they work per day must be inversely proportional. So we know that if there are 12 typists we need 4/3 times more hours than if we have 16 typists. Therefore we have to multiply by the number of hours that it takes 12 typists by 3/4 to solve. 

144 * 3/4 = 108

Now we know it takes 108 hours in total, and that means that 108/18 hours or 6 hours per day. 

Answer:

6 hours per day! :)

Also, note that instead of converting the number of hours per day to the total amount of hours, since the number of days they are working constant, we could've just kept it constant and ignore the number of hours they are working per day, so then it will be 8 * 3/4 which gives us 6 hours as previously predicted. 

50x + 1250 <= 12500

50x <= 11250

x <= 225

This means that each student has to raise at least 225 dollars!!

Enjoy!!

(Also, it would be great if someone told me to do the greater than/equal to sign because when I type it is <=)

Since the legs of the right triangle are 20 and 15, we can use the Pythagorean Theorem to find the hypotenuse which is the other side. 

a^2 + b^2 = c^2

15^2 + 20^2  = c^2

225 + 400 = c^2

625 = c^2

c = 25

Therefore, the hypotenuse of the right triangle or the final side is 25, now we can find the sum of the sides of the right triangle. 

25 + 20 + 15 = 60 

Since the square and right triangle have equal perimeters the square's perimeter has to be 60 too!

4a = 60

a = 15

One of the square's sides is 15, and it's area is simply side squared, which is 15^2 and that is 225. 

Answer: 

225 square inches

Enjoy!! 

Repeated roots occur when the quadratic can be factored into two or more binomials. 

For example....

(x^2 + 6x + 9) has repeated roots because it can be factored into (x + 3)(x + 3), which only give one distinct root which is -3. 

On the other hand...

(x^2 + 5x + 6) does not have repeated roots because it cannot be factored into a product of two of the same binomials and factors to (x + 2)(x + 3), which gives us too distinct roots which are -2, and -3. 

The first one doesn't factor evenly. 

The second one factor into 3(x-8)(x+7), which has two distinct roots which are 8, and -7. 

The third one doesn't factor evenly. 

The fourth one factors into (5x-3)(5x-3) which has repeating roots which are 3/5 and is the answer. 

The fifth one factors into (x-12)(x+2) which has two distinct roots which are 12 and -2. 

Therefore, our answer is the 4th quadratic which is 25x^2 - 30x + 9, which factorizes (5x-3)^2 and has one repeating root that is 3/5. 

Yeah, I knew that but accidentally misread the question, but I decided to post both for the viewer's knowledge. 

Or... Another way to solve it is this, we can make an equation. 

We know that the cheetah population is decreasing by 10% each year so each year it will be (100-10)% = (90% = 0.9) of the amount of cheetah's the previous year. Since it is going on for 11 years it's exponential. 

1750 * (0.9)^11 

= 549.17

Rounding to the nearest integer we get 549 cheetahs. 

Answer: 

About 549 cheetahs

To find the arc length you have to use the following formula: 

2*pi*r*(arcmeasure)/360

Let me explain this formula a bit...

2*pi*r, is the circumference of the circle, and we are finding a sector of that. 

The sector is the fraction (arcmeasure/360), which is how much it takes up of the circle.

In this case, the arcmeasure is 180, since it's a straight angle, so the following equation would be 

2*pi*r*(180/360)

= 2*1/2*pi*r

= pi*r

The radius isn't given, so if it was the arc length of the semicircle would be pi times the radius. 

Enjoy!! :) :)

To find the arc length you have to use the following formula: 

2*pi*r*(arcmeasure)/360

Let me explain this formula a bit...

2*pi*r, is the circumference of the circle, and we are finding a sector of that. 

The sector is the fraction (arcmeasure/360), which is how much it takes up of the circle.

In this case, the arcmeasure is 180, since it's a straight angle, so the following equation would be 

2*pi*r*(180/360)

= 2*1/2*pi*r

= pi*r

The radius isn't given, so if it was the arc length of the semicircle would be pi times the radius. 

Enjoy!! :) :)

Since the brick path is around the fountain, which is circular and has a diameter of 7 (radius: 3.5), we can find the circumference of the circle.

Circumference: pi * 2 * r

= pi * 2 * 3.5

= 7 pi

= 7 * 3.14

= 21.98

Answer: 

21.98 ft

I'm going to answer the solution in Point-Slope and Slope-Intercept. 

Slope - Intercept form is this: y = mx + b

We know that (x, y) is a point on the line, m is the slope, and b is the y-intercept. 

From the given information we have that it has a slope of 2 (m = 2), and one of the points on the line is (3, 9) which means that (x = 3, y = 9)

Plugging in we get:

y = mx + b

9 = 2*3 + b

9 = 6 + b

b = 3

Therefore the y-intercept is 3. After having this information we can construct our equation.

We now have to put this in the point-slope form which is: (y - y1) = m(x - x1)

Transferring what we had before we can plugin and get y - 9 = 2(x - 3)

Comment down if you have any queries or any questions on the solution. Tell me as soon as possible. Good luck in your Algebra Pursuit!! :)

Enjoy!! I put the answer in both point-slope and slope-intercept if you haven't noticed!! :)

So our answer is:

y - 9 = 2(x - 3)​

To solve this question, you have to make systems of equations, for each point. 

As you probably know, the equation of a circle is (x-h)^2 + (y-k)^2 = r^2, where (x, y) is a point on the circle, and (h, k) the center of the circle, and r is the radius. 

Now what you have to do is to plug in the values of x and y, and set them equal to each other since we know that the radius is constant. 

First of all, you can easily find out the diameter and the radius of the circle, because you have two points where the y-coordinates are constant, so then finding that you can do |-2 -2| which gives you 4, so now you know that the diameter is 4, and the radius is 2. 

From this plugging in (-2, 0) you get, 

(-2 - h)^2 + (0 - k)^2 = r^2

(-2 - h)^2 + (0 - k)^2 = 4

I'm sure that after you have the idea from here you can distribute, and make 2 more similar equations, then equate and solve. Comment down, if you need another hint or another part of the solution!! 

Good Luck with your Geometry Passion!! :)

:) 

Let's say that the length is x and the width is y, then the perimeter is 2x + 2y, and making that into an equation we get...

2x + 2y = 56

Simplifiying we get...

x + y = 28

The ratio of the length and width is 4 to 3, so we can make the equation...

4x = 3y

3(x + y) = 28

3x + 3y = 84

Plugging in 3y as 4x we get 

3x + 4x = 84

7x = 84

x = 12

Now we plug x = 12, into the original equation:

x + y = 28

x = 12

y = 28 - 12 = 16

The question asks us to find the diagonal, and that is going to be using the Pythagorean theorem. 

In this case, the hypotenuse is the diagonal of the triangle if we split the rectangle along the diagonal. 

a^2 + b^2 = c^2

12^2 + 16^2 = 400

c^2 = 400

c = 20

Since we know that the hypotenuse (c) is diagonal, and that is what the question is asking for it is our answer.

Answer: 20 meters