The slope of a roof is called the pitch and is defined as follows:
pitch = rise of roof
½ span of roof
Find the pitch of a roof if the rise is 12 feet and the span is 30 feet.
Find the pitch of a roof if the rise is 4 feet and the span is 24 feet.

Solve for x in 5x - 3y = 18

x = 3(6 + y)/5

Substitute x = 3(6 + y)/5 into 3x + 3y = 30

9(6 + y)/5 + 3y = 30

Solve for y in 9(6 + y)/5 + 3y = 30

y = 4

Substitute y = 4 into x = 3(6 + y)/5

x = 6

Therefore,

x = 6

y = 4

The solution to the system of equations is x = 6 and y = 4.

Step 1: Add the two equations.

5x - 3y = 18

3x + 3y = 30

8x = 48

Step 2: Solve for x.

8x = 48

x = 6

Step 3: Substitute x = 6 into one of the original equations to solve for y.

5(6) - 3y = 18

30 - 3y = 18

-3y = -12

y = 4

Answer:

both are

Step-by-step explanation:

A number, or a polynomial, is "divisible" by a "factor." Since (k+5) is a factor of (k^2-25), (k^2-25) is divisible by (k+5). Both Eleanor and Isabel are correct in their description.

Answer:

The answer A

Step-by-step explanation:

B: You need a side that is longer than 2 equal sides. B is incorrect.

A: Try

9^2 + 12^2 = 15^2 if that works, A will work.

81 + 144 = 225

This works. I used these numbers because they were easier to see.

0.9^2 + 1.2^2 = 1.5^2

0.81 + 1.44 = 2.25

2.25 = 2.25

That is the answer A

But we have to check the  other two.

C:

0.9^2 = 0.81

1.2^2 = 1.44

1.8^2 = 3.24   This won't work. 3.24 is just too big.

D:

0.9^2 = 0.81

0.6^2 = 0.36

1.5^2 = 2.25   It's too far away.

When dealing with numbers, we find the LCM by factoring the numbers, and then choosing all the primes appearing, with the highest exponent possible.

Similarly, we will factor the polynomials first:

[tex] x^4+3x^3 = x^3(x+3)[/tex]

[tex]x^2-9 = (x+3)(x-3)[/tex]

[tex]x^3+6x^2+9x = x(x^2+6x+9) = x(x+3)^2[/tex]

The following factors emerged:

So, the LCM is

[tex]x^3(x-3)(x+3)^2[/tex]

I’m pretty sure it’s A

Answer: A

Step-by-step explanation:

Answer:

y=3/4 *x

Step-by-step explanation:

1. the given equation is the equation of line in slope-interception form (y=kx+b), where 3/4 is 'k' and (-4) is 'b'.

2. if the system of the two equations has no solution, it means, the 2d line is parallel to the given one, then 'k' of the 2d line is the same value.

3. common view of the equation of the 2d line is y=3/4 *x+b, where 'b' is number. For example, y=3/4x; y=3/4 *x+1; y=3/4 *x-5/8, etc.

One example of a linear function that generates a system without a solution is given by:

[tex]y = \frac{3}{4}x + 5[/tex]

A linear function is modeled by:

y = mx + b

In which:

If two equations have the same slope but different intercepts, they never  intersect each other, that is, a system without solutions is generated. Hence, one possible example in this problem is given by:

[tex]y = \frac{3}{4}x + 5[/tex]

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Answer:

perimeter = (18 +9π) cm

area = (81 -20.25π) cm^2

Step-by-step explanation:

The perimeter of the shaded area is the circumference of the circle added to two sides of the square. The circumference of the circle is π times the diameter, so the perimeter is ...

p = 2(9 cm) + π(9 cm) = (18 +9π) cm

___

The area of the shaded portion is the difference between the area of the square and the area of the circle. The area of the square is the square of the diameter. The area of the circle is π/4 times that value.

A = (9 cm^2) + (π/4)(9 cm^2) = (81 +20.25π) cm^2

_____

Comment on circle area

The formula you often see is ...

A = πr^2 . . . . r is the radius

since r = d/2, where d is the diameter, this can also be written as ...

A = π(d/2)^2 = (π/4)d^2

Here, the diameter of the circle is the same as the side length of its enclosing square, so the area of the circle is π/4 times the area of the enclosing square.

Answer:

Area= [tex]81-\frac{81}{4}\pi[/tex]

Perimeter= [tex]9\pi +18[/tex]

Answer:

1. y = 4n -1

2. y = 9n -2

Step-by-step explanation:

1. For some integer n, numbers that are congruent to 2 mod 4 are ... 4n+2. Then you have ...

y + 3 = 4n +2 . . . . next, subtract 3 from both sides

y = 4n -1 . . . . values of y are natural numbers for all natural numbers n

___

2. Similarly, ...

3 - y = 9n +5

3 - 9n - 5 = y . . . . . add y-9n-5

-9n -2 = y . . . . . . . . collect terms

For every integer n, there is also an integer -n. For our purpose, we're only interested in those values of n that make -9n be positive. Rather than require n be negative so -9n is positive, we can require n be positive and use the expression 9n for the multiple of 9.

We want y a natural number, so we can write this as ...

y = 9n -2 . . . . for natural numbers n

To find the natural number solutions for congruences, we use algebraic manipulation to simplify the equation and find values that satisfy the congruence. For the given congruences, the natural number solutions are y = 3, 7, 11, 15, ... and y = 7, 16, 25, ..., respectively.

1. To find the natural number solutions for the congruence y + 3 ≡ 2 mod 4, we need to find all values of y that satisfy the given congruence.

Subtracting 3 from both sides of the congruence, we have y ≡ -1 mod 4. Since we are looking for natural number solutions, we can rewrite this as y ≡ 3 mod 4.

The natural number solutions for this congruence are y = 3, 7, 11, 15, ...

2. For the congruence 3 - y ≡ 5 mod 9, we need to determine the natural number solutions for y that make the congruence true.

Subtracting 3 from both sides, we have -y ≡ 2 mod 9. Multiplying both sides by -1, we get y ≡ -2 mod 9. Rewriting this as y ≡ 7 mod 9, the natural number solutions for y are y = 7, 16, 25, ...

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Seven thousand sixty-five fifty six

Answer:

seven hundred sixty-five thousand fifty-six

Step-by-step explanation:

Your Welcome

Answer:

Largest Median: Same

Largest Range: Castro

Largest IQR: Castro

Step-by-step explanation:

With a box-and-whisker plot, the box represents the upper and lower quartiles, the vertical line inside the box represents the median, and the lines on either side of the box show the high and low of the range.

Largest Median: Medians are the same because the verticle line inside the boxes is at 7 for both

Largest Range: Ms Castro's Class- the lines on either side of the box for Ms Castro go from 1-10 while the other class only goes from 4-10.

Largest IQR (interquartile range) Ms. Castro's class: their IQR goes from 5-8 while the other class only goes from 6-8

Answer:

same median and CASTRO CASTRO just took the TYS on algebra nation

Step-by-step explanation:

Answer:

The difference is that the simple interest gives you interest only on the deposited amount in the account while the compound interest is calculated over the deposited amount and the previous interest amount added to the deposited amount.

Step-by-step explanation:

The difference between simple and compound interest is that simple interest is calculated only on the principal amount, while compound interest is earned on past interest.

The difference between simple and compound interest in a savings account is that simple interest is an interest rate calculation only on the principal amount, while compound interest is interest that is earned on past interest. This means that with compound interest, the total amount of savings will grow dramatically over time. To illustrate this, consider the example of putting $1,000 in a savings account with 10% interest. With simple interest, after one year, you will have $1,100. But with compound interest, if you leave the $1,100 in the account and continue to earn 10% interest, you will have $1,210 at the end of year two.

Answer:

60

Step-by-step explanation:

∑[i=1,5] 4i = 4·1 + 4·2 + 4·3 + 4·4 + 4·5

= 4 + 8 + 12 + 16 + 20

= 60

Answer:

a) (0.08)^6

Step-by-step explanation:

Given in the question,

[tex]\sqrt{0.08^{12} }[/tex]

As we know that

[tex]\sqrt{x} = x^{1/2}[/tex]

So,

[tex]\sqrt{0.08^{12}[/tex] = [tex]0.08^{12(1/2)}[/tex]

[tex]0.08^{12/2}[/tex]

[tex]0.08^{6}[/tex]

Answer:

  A.  216 cm³

Step-by-step explanation:

The volume of a cube is the cube of the side length:

  V = s³ = (6 cm)³ = 216 cm³

Answer:

C. 6.683

Step-by-step explanation:

Answer:

3 - 2√5

-------------  

      2

Step-by-step explanation:

These two expressions have different denominators, and thus use of the LCD is necessary.  The LCD is 4.

Thus, we have:

6/4 - (4/4)√5, or

6 - 4√5

-------------

      4

This can be reduced to:

3 - 2√5

-------------   (answer)

      2

a cube would be formed as ABCD form a square when graphed

Answer:

Option D. minimum value at −38

Step-by-step explanation:

we have

[tex]x^{2}-12x-2[/tex]

Let

[tex]y=x^{2}-12x-2[/tex]

Complete the square

[tex]y+2=x^{2}-12x[/tex]

[tex]y+2+36=(x^{2}-12x+36)[/tex]

[tex]y+38=(x^{2}-12x+36)[/tex]

[tex]y+38=(x-6)^{2}[/tex]

[tex]y=(x-6)^{2}-38[/tex] ------> equation of a vertical parabola in vertex form

The vertex is the point [tex](6,-38)[/tex]

The parabola open upward-----> the vertex is a minimum

therefore

minimum value at −38

Answer:

4

Step-by-step explanation:

Let b represent the number of birds. Then the cost of care for b birds and 16-b cats was ...

4.50b +6.50(16-b) = 96.00

-2.00b = -8.00 . . . . . . . . . . . . simplify, subtract 104

b = 4 . . . . . . . . . . . . . . . . . . . . .divide by -2

The number of birds at the shelter on Thursday was 4.

Let

[tex]\theta=\sin^{-1}\dfrac x2\implies\sin\theta=\dfrac x2[/tex]

Recall that

[tex]\cos\theta=\sqrt{1-\sin^2\theta}=\sqrt{1-\dfrac{x^2}4}[/tex]

Then

[tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}\implies\tan\left(\sin^{-1}\dfrac x2\right)=\dfrac{\frac x2}{\sqrt{1-\frac{x^2}4}}[/tex]

[tex]\implies\tan\left(\sin^{-1}\dfrac x2\right)=\dfrac x{\sqrt{4-x^2}}[/tex]

Final answer:

To find tan(sin⁻¹(x/2)), we use the relationship between the sides of a right triangle, leading to the solution tan(θ) = x/√(4 - x²).

Explanation:

The question "what is tan(sin-1(x/2))" involves understanding inverse trigonometric functions and their properties. To solve this, we can create a right-angled triangle where the angle θ has a sine value of x/2. According to the Pythagorean theorem, if one side (opposite) is x and the hypotenuse is 2, the adjacent side can be calculated as √(4 - x²). Thus, tan(θ) = opposite/adjacent = x/√(4 - x²).

Answer:

a = 12

Step-by-step explanation:

Multiply the equation by the inverse of the coefficient of "a".

4·(1/4)a = 4·3

1·a = 12 . . . . . . simplify

a = 12

When writing rational expressions, you need to be aware that ...

  1/4x = (1/4)x ≠ 1/(4x)

Parentheses around the denominator are required, unless you're typesetting the expression and can use a fraction bar for grouping.

The derivative of the curve expression is ...

  y' = x - 1/(4x) . . . . . parentheses added to what you wrote

and the expression (1 -(y')^2) can be written ...

  1 -(y')^2 = x^2 +1/2 +1/(16x^2) . . . . . parentheses added to what you wrote

The first and last terms of this trinomial are both perfect squares, so you might suspect the whole trinomial is a perfect square. You recall that ...

  (a +b)^2 = a^2 + 2ab + b^2

This is a good "pattern" to remember. Using it is a matter of pattern recognition, as is the case with a lot of math.

Here, you have ...

  a = x

  b = 1/(4x)

In order for your trinomial to be a perfect square, the product 2ab must equal the middle term of your trinomial. (Spoiler: it does.)

  2ab = 2(x)(1/(4x)) = (2x)/(4x) = 1/2 . . . . . matches the middle term of 1 -(y')^2

Hence your trinomial can be written as the square ...

  1 -(y')^2 = (x +1/(4x))^2

_____

This is convenient because you want to integrate the square root of this. Your integral then becomes ...

[tex]\displaystyle\int\limits_{2}^{4}{\left(x+\frac{1}{4x}\right)\,dx[/tex]

Answer:

48.3 cm^2

Step-by-step explanation:

The area of the enclosing square is that of a square 15 cm on a side, so is ...

square area = (15 cm)^2 = 225 cm^2

The white area is that of a quarter-circle of radius 15 cm, so that area is ...

quarter-circle area = (1/4)πr^2 = (π/4)·(15 cm)^2 = 56.25π cm^2 ≈ 176.7 cm^2

Then the yellow area is the difference ...

yellow area = square area - quarter-circle area

= (225 cm^2) - (176.7 cm^2)

yellow area = 48.3 cm^2

Answer:

about 0.08087

Step-by-step explanation:

The statistics functions of your graphing calclulator can help you with that. It is a good idea to learn to use them.

Answer:

Part A:  A (6 , 11) , B (5 , 6) , C (7 , 1) , D (0 , 8)

Part B:  A (-6 , -11) , B (-5 , -6) , C (-7 , -1) , D (0 , -8)

Step-by-step explanation:

* Lets study the reflection about the two axes X and Y

- The distance between the point and the axes of reflection =

 the distance between its image and the axes

- The point and the its image are on opposite sides of the axes

- If a point (x , y) reflected about x axis, that means the point

  will move vertically

- Moving vertically means we will change the sign of the y-coordinates

∴ The image of (x , y) after reflection about x-axis is (x , -y)

- If a point (x , y) reflected about y axis, that means the point

  will move horizontally

- Moving horizontally means we will change the sign of the x-coordinates

∴ The image of (x , y) after reflection about x-axis is (-x , y)

* Now lets use the explanation above to solve our problem

- At first lets right the original point of the quadrilateral ABCD

∵ A (-6 , 11) , B (-5 , 6) , C (-7 , 1) , D (0 , 8)

Part A: The y-axis is the line of reflection

- Lets change the signs of x-coordinates in all points

∴ The new points after reflection about y-axis is:

  A (6 , 11) , B (5 , 6) , C (7 , 1) , D (0 , 8)

- Note: The point D does not change because x-coordinate is 0

            and there is no sign for the 0

Part B: The x-axis is the line of reflection

- Lets change the signs of y-coordinates in all points

∴ The new points after reflection about x-axis is:

  A (-6 , -11) , B (-5 , -6) , C (-7 , -1) , D (0 , -8)

Answer:

The equation of the line is y = 3/4x - 1/4

Step-by-step explanation:

To find the equation of this line, start by using the two points with the slope formula to find the slope.

m(slope) = (y2 - y1)/(x2 - x1)

m = (2 - 1/2)/(3 - 1)

m = (3/2)/2

m = 3/4

Now that we have the slope, we can use that and either point in point-slope form to find the equation.

y - y1 = m(x - x1)

y - 2 = 3/4(x - 3)

y - 2 = 3/4x - 9/4

y = 3/4x - 1/4

Answer:

58 at the point (9,8)

7 at the point (1, 1)

Step-by-step explanation:

The maximum points will be found in the vertices of the region.

Therefore the first step to solve the problem is to identify through the graph, the vertices of the figure.

The vertices found are:

(1, 10)

(1, 1)

(9, 5)

(9, 8)

We look for the values of x and y belonging to the region, which maximize the objective function [tex]f(x, y) = 2x + 5y[/tex]. Therefore we look for the vertices with the values of x and y higher.

(1, 10), (9, 5), (9, 8)

Now we substitute these points in the objective function and select the one that produces the highest value for f (x, y)

[tex]f(1, 10) = 2(1) + 5(10) = 52\\\\f(9, 5) = 2(9) + 5(5) = 43\\\\f(9, 8) = 2(9) + 5(8) = 58[/tex]

The point that maximizes the function is:

[tex](9, 8)[/tex] with [tex]f(9, 8) = 58[/tex]

Then the value that produces the minimum of f(x, y) is (1, 1)

[tex]f(1, 1) = 2(1) + 5(1) = 7[/tex]

Compare [tex]\dfrac1{\sqrt{n^2+1}}[/tex] to [tex]\dfrac1{\sqrt{n^2}}=\dfrac1n[/tex]. Then in applying the LCT, we have

[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac1{\sqrt{n^2+1}}}{\frac1n}\right|=\lim_{n\to\infty}\frac n{\sqrt{n^2+1}}=1[/tex]

Because this limit is finite, both

[tex]\displaystyle\sum_{n=1}^\infty\frac1{\sqrt{n^2+1}}[/tex]

and

[tex]\displaystyle\sum_{n=1}^\infty\frac1n[/tex]

behave the same way. The second series diverges, so

[tex]\displaystyle\sum_{n=0}^\infty\frac1{\sqrt{n^2+1}}=1+\sum_{n=1}^\infty\frac1n[/tex]

is divergent.

The limit comparison test is used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series.

The limit comparison test is used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series. The test states that if the limit of the ratio between the terms of the given series and the terms of a known convergent series is a finite positive number, then both series behave in the same way. Here is how you can use the limit comparison test step by step:

For example, if we have the series ∑(n^2)/(2^n), we can use the limit comparison test with the series ∑(1)/(2^n). Taking the limit of the ratio (n^2)/(2^n) / (1)/(2^n) as n approaches infinity, we get:

lim(n→∞) [(n^2)/(2^n)] / [(1)/(2^n)] = lim(n→∞) (n^2)/(1) = ∞

Since the limit is infinite, the result is inconclusive. Therefore, the given series does not converge or diverge using the limit comparison test.

Final answer:

To plot a graph of the amount Betsy earns against the time she works, we'll put the hours worked on the x-axis and the amount earned on the y-axis. Betsy is paid $11.50 per hour and the equation for Betsy's earnings is Y = 11.50X.

Explanation:

To plot a graph of the amount Betsy earns against the time she works, we'll put the hours worked on the x-axis and the amount earned on the y-axis. In this case, Betsy worked an 8 hour shift and was paid $92.00. So, the point on the graph would be (8, 92). To find how much Betsy is paid per hour, we can divide the total amount earned by the number of hours worked. In this case, $92.00 divided by 8 hours equals $11.50 per hour.

Using the equation Y = MX + B, where Y is the amount earned, M is the rate per hour, X is the number of hours worked, and B is the y-intercept, we can write the equation for Betsy as Y = 11.50X + 0, since she doesn't have a one-time fee.