Mathematics College

Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box. Also, specify any restrictions on the variable.a²-36/5a + 30Rational expression in lowest terms:Variable restrictions for the original expression: a

Answers

Answer 1

The expression is given to be:

[tex]\frac{a^2-36}{5a+30}[/tex]

From the numerator, using the difference of two squares, we have:

[tex]a^2-36=a^2-6^2=(a-6)(a+6)[/tex]

From the denominator, by factorization, we have:

[tex]5a+30=5(a+6)[/tex]

Therefore, the expression becomes:

[tex]\Rightarrow\frac{(a-6)(a+6)}{5(a+6)}[/tex]

Cancel out common terms in the denominator and numerator. The simplified expression will be:

[tex]\Rightarrow\frac{a-6}{5}[/tex]

From the original expression, the variable restriction will be at:

[tex]\begin{gathered} 5a+30=0 \\ Solving \\ 5a=-30 \\ a=-\frac{30}{5} \\ a=-6 \end{gathered}[/tex]

The restriction is:

[tex]-6[/tex]

Related Questions

Please help me solve this problem it’s my last question on my hw

Answers

To solve this problem, we will use the following expression:

[tex]\frac{favorable\text{ outcomes}}{total\text{ outcomes}}.[/tex]

In this case:

[tex]\begin{gathered} Total\text{ outcomes}=21+6+6=33, \\ favorable\text{ outcomes}=6. \end{gathered}[/tex]

Substituting the above values in the expression, we get:

[tex]\frac{6}{33}.[/tex]

Finally, simplifying the above result, we get that the probability that his shirt number is from 67 to99 given that he weighs at most 210 pounds is:

[tex]\frac{2}{11}.[/tex]

Answer:

[tex]\frac{2}{11}.[/tex]

write the quadratic equation whose roots are -2 and 1 and whose leading coefficient is 4 using the letter x to represent the variable

Answers

The quadratic equation is y = 4. (x + 2)(x - 1).

Given,

The roots are -2 and 1 and whose leading coefficient is 4 using the letter x to represent the variable.

To Write the Quadratic Equation by using above given:
Now, According to the question:

Roots are -2 and 1

Leading Coefficient is = 4

Using Variable is = x

Formulate the equation of polynomial function is

y = 4. (x + 2)(x - 1)

Hence, The quadratic equation is y = 4. (x + 2)(x - 1).

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price. Find the sale price. There are 252 students on the student council at West High School. If there are 700 students enrolled, what percent are on the student council?

Answers

We have in total 700 students

252 students are on the students council

700 is 100%

252 is x

x represents the percentage of the student's council, in order to calculate x we need to do the next operation

[tex]x=\frac{252\cdot1}{700}=0.36[/tex]

252 is the 36%

the percent of students that are on the student council is 36%

A candy company has decided to sell candy in a new package that is shaped like a triangular prism in order to calculate the volume of the package, the area of the base must be determined.If the base of the package is equilateral triangle with sides 10 centimeters (cm) long, what is the area of the base?

Answers

Using the formula for the area of the equilateral triangle, we have:

[tex]\begin{gathered} A=\frac{\sqrt[]{3}}{4}a^2 \\ A=\frac{\sqrt[]{3}}{4}(10)^2cm^2\text{ (Replacing)} \\ A=\frac{\sqrt[]{3}}{4}(100)^{}cm^2\text{ (Raising 10 to the power of 2)} \\ A=25\sqrt[]{3}\text{ }cm^2\text{(Dividing 100 by 4}) \\ A=43.3\text{ }cm^2\text{ (Multiplying 25 by }\sqrt[]{3}) \\ \text{The answer is 43.3 cm}^2 \end{gathered}[/tex]

help me with the 17 pls just give me the answer

Answers

Solution

Evaluate each function below for the given equation

17.

[tex]f(x)=11x-9[/tex]

f(2) is when the value of x = 2

[tex]\begin{gathered} f(x)=11x-9 \\ f(2)=11(2)-9 \\ f(2)=22-9 \\ f(2)=13 \end{gathered}[/tex]

18.

[tex]g(x)=-4(g+2)\text{ for g\lparen5\rparen}[/tex]

g(5) is when the value of x = 5

[tex]\begin{gathered} g(x)=-4(g+2) \\ g(x)=-4g-8 \\ g+4g=-8 \\ 5g=-8 \\ g=-\frac{8}{5} \end{gathered}[/tex]

19.

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

f(-1) is when the value of x = -1

[tex]\begin{gathered} f(x)=3+2(-x)\text{ for x = -1} \\ f(-1)=3+2(-(-1)) \\ f(-1)=3+2 \\ =5 \end{gathered}[/tex]

Pls help asap

Find the product of 5.3 x 10^12 and 5.12 x 10^9. Write the final answer in scientific notation.

2.7136 x 10^21

2.7136 x 10^22

27.136 x 10^21

27.136 x 10^22

Answers

Answer:

B. 2.7136 x 10^22

Step-by-step explanation:

Hope this helps :))

The dimensions of a tennis court are 27 feet by 78 feet. The dimensions of a ping pong table are 9 feet by 5 feet. Is a tennis court a dilation of a ping pong table? Explain why or why not.

Correctly answer yes or no (4 points)
Provide a clear and logical explanation (6 points)

Answers

No, The ratios are not equal tennis court is not a dilation of the table.

Given,

The dimensions of a tennis court are 27 feet by 78 feet.

The dimensions of a ping pong table are 9 feet by 5 feet.

To check the:

Is a tennis court a dilation of a ping pong table?

Now, According to the question:

For Tennis Court :

Dimensions is 27feet by 78feet

For Ping Pong Table:

Dimensions is 9 feet by 5 feet

To check that:

27/5 = 27/5 (Take lengths of tennis court and ping pong table )

78/9 = 26/3 (Take breadths of tennis court and ping pong table )

∴ 27/3 ≠ 26/3

Since, No, The ratios are not equal tennis court is not a dilation of the table.

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Which table represents a linear function?

Answers

The most appropriate choice for function will be given by-

The first option is correct

What is a function?

A function from A to B is a rule that assigns to each element of A a unique element of B. A is called the domain of the function and B is called the codomain of the function.

There are different operations on functions like addition, subtraction, multiplication, division and composition of functions.

Here,

For the first table,

Slope of function from x = 1 to x = 2

= [tex]\frac{1 - \frac{1}{2}}{2 - 1}[/tex]

=[tex]\frac{1}{2}[/tex]

Slope of function from x = 2 to x = 3

= [tex]\frac{\frac{3}{2}-1}{3 - 2}[/tex]

= [tex]\frac{1}{2}[/tex]

Slope of function from x = 3 to x = 4

= [tex]\frac{2-\frac{3}{2}}{4 - 3}[/tex]

= [tex]\frac{1}{2}[/tex]

Here the slope is same. So the function is linear

For the second table,

Slope of function from x = 1 to x = 2

= [tex]\frac{\frac{1}{2} - 1}{2 - 1}[/tex]

=[tex]-\frac{1}{2}[/tex]

Slope of function from x = 2 to x = 3

= [tex]\frac{\frac{1}{3}-\frac{1}{2}}{3 - 2}[/tex]

= [tex]-\frac{1}{6}[/tex]

Here the slope is not same. So the function is not linear

For the third table,

Slope of function from x = 1 to x = 2

= [tex]\frac{9 - 7}{2 - 1}[/tex]

= 2

Slope of function from x = 2 to x = 3

= [tex]\frac{13 - 9}{3 - 2}[/tex]

= 4

Here the slope is not same. So the function is not linear

For the Fourth table,

Slope of function from x = 1 to x = 2

= [tex]\frac{6 - 0}{2 - 1}[/tex]

=[tex]6[/tex]

Slope of function from x = 2 to x = 3

= [tex]\frac{16 - 6}{3 - 2}[/tex]

= 10

Here the slope is not same. So the function is not linear

So the first option is correct

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Write in point-slope form an equation of the line through the pair of points (2,0) and (6,10). Type an equation of the line in point-slope form using one of the given points.

Answers

The point-slope form of a line is

[tex]y-y_1=m(x-x_1)_{}[/tex]

where (x_1, y_1) are the coordinates of a point on the line.

From the given point we find the slope m.

[tex]m=\frac{10-0}{6-2}=2.5[/tex]

With the value of the slope in hand, we now use one of the points given to write down the point-slope form:

[tex]\textcolor{#FF7968}{y-10=2.5(x-6)}[/tex]

where we have used the point (6, 10).

a. Write a formula for the perimeter P of the square.Type your answer into the box below.b. Write a formula for the area A of the square,Type your formula into the box below.

Answers

a. The formula for the perimeter of a square is P=4L, where L is the length of it's sides.

b. The formula for the area of a square is A=LxL=L^2, where L is the length of it's sides.

If 9x - 3 = 4, what does 9x - 2 = ?

Answers

it is given that

9x - 3 = 4

then we can write this

9x - (2+1) = 4

9x - 2 -1 =4

9x - 2 = 4 +1

9x - 2 = 5

so the answer is 5

How do i fill the boundaries for each class? how do i draw a histogram to illustrate this information ? how can i calculate the median age of the population?

Answers

Given:

The tabular representation of the data is given.

Required:

Histogram and Median of the given data.

Explanation:

The histogram of the given data is calculated as follows:

Figure 1

The median of the given set of data is calculated as,

Answer:

Thus figure 1 represents the histogram of the given data. The median of the given set of data is 27.7353.

convert 450 m/s to m/h

Answers

1,620,000 m/h

1) To convert 450 m/s we can set out the following:

[tex]\frac{450}{1s}\text{ }\times3600s=1,620,000[/tex]

Since 1 hour is equivalent to 3600s then we can simply multiply by 3600 once the unit meter remains the same. m/s to m/h

2) Hence, the answer is 1,620,000 m/h

grant is rolling a six sided number die eight times. How many possible outcomes are there?

Answers

The die has 6 faces

[tex]\begin{gathered} \text{ In n number of throws, the number outcomes is given by 6}^n \\ ^{} \end{gathered}[/tex][tex]\begin{gathered} \text{ In 8 throws, the possible outcomes will be 6}^8 \\ =1679616 \end{gathered}[/tex]

Evaluate each function in the table at f (3).
Enter the correct answers in the boxes in the table.
Σ
Functions
Function
f(x) = 3x + 5
f(x)=x²-1.5
3
f(x) = 2x
ƒ(3)

Answers

The answer to the given functions are:

1) f (3) = 3 × 3 + 5 = 14

2) f( 3 ) = (3)² - 1.5 = 7.5

3) f (3) = 3×2 = 6

What are functions ?

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.

A function is a type of rule that produces one output for a single input.. This is illustrated by the equation y=x2. Any input for x results in a single output for y. Considering that x is the input value, we would state that y is a function of x.

Given : Functions are f(x) = 3x + 5 ------- 1

f(x)=x²-1.5 --------- 2 and f(x) = 2x -----------3

Thus when we put x = 3 we will get the following values that is ;

1 becomes f (3) = 3 × 3 + 5 = 14

and 2 becomes f( 3 ) = (3)² - 1.5 = 7.5

And 3 becomes f (3) = 3×2 = 6

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Assume that adults have IQ scores that are normally distributed with a mean of μ=100 and a standard deviation σ=20. Find the probability that a randomly selected adult has an IQ less than 128.

Answers

The probability that the IQ is less than 128 is 0.91924

How to determine the probability?

The given parameters about the distribution are

Mean value of the set of data, μ = 100Standard deviation value of the set of data σ = 20IQ value, x = 128

The z-score of the data value is calculated as

z = (x - μ)/σ

Substitute the given parameters in the above equation

z = (128 - 100)/20

Evaluate

z = 1.4

The probability that the IQ is less than 128 is then calculated as:

P(x < 128) = P(z < 1.4)

From the z table of probabilities, we have;

P(x < 128) = 0.91924

Hence, the probability is 0.91924

Answers

Since there is no overlapping the area of the polygon must be equal to the combined area of the cards.

Find the orthogonal trajectories of the family of curves. y^2=6KX^3

Answers

Y4 =9KX is your answer

The orthogonal trajectories of the family of curves y² = 6Kx³ will be [tex]y = \sqrt{6(C-\frac{x^2}{4}}[/tex]

What is the orthogonal trajectory?

An orthogonal trajectory of a family of curves is any curve that forms right angles with each member of the family. The curve does not always have to cross each member of the family.

It is given that,

y² = 6Kx³

Differentiate both sides we get,

[tex]\rm 2y = 6k(3x^2) \frac{dy}{dx} \\\\ \frac{dy}{dx} =\frac{2y}{18Kx^2}[/tex]

Substitute the value of K as

[tex]\rm K = \frac{y^2}{6x^3}[/tex]

[tex]\frac{2y}{18(\frac{y^2}{6x^3})x^2} = \frac{dy}{dx} \\\\ \frac{dy}{dx} = \frac{2x}{3y}[/tex]

Consider the negative reciprocal to find the perpendicular slope.

[tex]\frac{-3y}{2x} =\frac{dy}{dx} \\\\ - \int {\frac{1}{2x} \ dx = \int \frac{1}{3y} dy \\\\[/tex]

[tex]-\frac{x^2}{4}+C = \frac{y^2}{6}[/tex]

Simplify,

[tex]\rm 6(-\frac{x^2}{4}+C)= y^2 \\\\ y = \sqrt{6(C-\frac{x^2}{4}}[/tex]

Thus, the orthogonal trajectories of the family of curves y² = 6Kx³ will be [tex]y = \sqrt{6(C-\frac{x^2}{4}}[/tex]

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The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.48 millimeters and a standard deviation of 0.07 millimeters. Find the two diameters that separate the top 3% and the bottom 3%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

Answers

The two diameters that separate the top 35 from the bottom 35 in the normal distribution are 5.35 mm and 5.61 mm .

Normal distributions are crucial to statistics because they are widely used in the natural and social sciences to represent real-valued covariates whose distributions are unknown.

Some of their significance comes from the main limit theorem. This claim states that, in some cases, the average of many samples (observations) of a stochastic process with limited mean and variance constitutes itself as a random variable, whose distribution tends to become more normal as the number of samples increases. Because of this, the distributions of physical quantities, like misspecification, which are thought to be the consequence of hundreds of distinct processes, are frequently close to normal.

First we will find the bottom 3% such that P(X ≤ x) = 0.03

⇒ [tex]P(\frac{X-5.48}{0.07}\leq \frac{x-5.48}{0.07} )=0.03[/tex]

⇒[tex]P(Z\leq \frac{x-5.48}{0.07} )=0.03[/tex]

Now we will use the normal table to calculate the corresponding z-score.

[tex]\frac{x-5.48}{0.07} =-1.88\\\\\implies x = 5.348[/tex]

Now we will find the same for the top part of the distribution.

[tex]P(\frac{X-5.48}{0.07}\leq \frac{x-5.48}{0.07} )=0.97\\\\\implies P(Z\leq \frac{x-5.48}{0.07} )=0.97[/tex]

Now we will use the normal table to calculate the corresponding z-score.

[tex]\frac{x-5.48}{0.07} =1.88\\\\\implies x = 5.612[/tex]

The two diameters that separate the top 35 from the bottom 35 in the normal distribution are 5.35 mm and 5.61 mm .

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TRIG ANALYSIS- THE LAW OF SINES- THE AMBIGUOUS CASERecall that your triangle should be set up so that angside b, and angle C is opposite side c.PLEASE HELP. WILL MARK BRAINLIEST

Answers

The sine law is described by the equation

[tex]\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}[/tex]

1) Given angle C, length of side b, and length of side c, we are looking for the value of angle B. The angle B can be computed using

[tex]\begin{gathered} \frac{\sin B}{b}=\frac{\sin C}{c} \\ \sin B=\frac{b}{c}\sin C \end{gathered}[/tex]