Evaluate The Rational Expression For The Given X Value. Express The Answer As A Fraction In Simplest (2024)

Statement Problem: Solve for the missing sides of the triangle;

Solution:

The sum of angles in a triangle is 180degrees. Thus,

[tex]\begin{gathered} \angle A+\angle B+\angle C=180^o \\ \angle B=180^o-\angle A-\angle C \\ \angle B=180^o-42^o-96^o \\ \angle B=42^o \end{gathered}[/tex]

Since measure angle A and measure angle B are equal. Thus, the triangle is isosceles and the two sides are equal.

[tex]a=b[/tex]

We would apply sine rule to find the missing side a.

[tex]\begin{gathered} \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \\ \frac{\sin A}{a}=\frac{\sin C}{c} \end{gathered}[/tex][tex]\begin{gathered} \frac{\sin42^o}{a}=\frac{\sin96^o}{12} \\ a=\frac{12\sin42^o}{\sin96^o} \\ a=8.07 \\ a\approx8.1 \end{gathered}[/tex]

Thus,

[tex]a=b=8.1[/tex]

CORRECT ANSWERS:

[tex]\begin{gathered} a=8.1 \\ b=8.1 \\ m\angle B=42^o \end{gathered}[/tex]

a) Since the cost of a bag of popcorn is $6 and the cost of a drink is $4,

[tex]\begin{gathered} T=6\cdot5+7\cdot4 \\ \Rightarrow T=30+28=58 \\ \Rightarrow T=58 \end{gathered}[/tex]

Therefore, the answer to the first question is $58.

b) Substitute 5 for p and 7 for d in the expression above; therefore,

[tex]T=6p+4d[/tex]

The total cost is given by the equation T=6p+4d, where T is in dollars, p is the number of bags of popcorn and d is the number of drinks.

a) f(0) = -1

b) f(1) = 1

c) f(4) = 7

d) f(5) = 121

Explanation:

. Since for every value between -2 (excluded) and 4 (included)

~ 0 , 1 and 4

You have to use the first equation

=> f(0) = 2 * 0 - 1 = -1

=> f(1) = 2 * 1 - 1 = 1

=> f(4) = 2 * 4 - 1 = 7

. For values between 4 (exclude) and 5(included)

~ 5

You have to use the second equation

=> f(5) = 5^3 - 4 = 121

Let:

x = Number of sodas purchased

y = Number of hamburgers purchased

The food truck charges $3 for sodas, so the total cost for sodas will be:

3*x=3x

also, it charges $8 for each hamburger, hence, the total cost for hamburgers will be:

8*y = 8y

Since Jack wants to spend no more than $30, the total cost must be less or equal than $30:

[tex]\begin{gathered} \text{Total cost }\leq\text{ 30} \\ \text{Total cost = total cost for sodas+total cost for hamburgers} \\ 3x+8y\le30 \end{gathered}[/tex]

Volume of a cylinder:

[tex]V=h*r^2*\pi[/tex]

For cylinder A:

[tex]10\pi cm^3=h*r^2*\pi[/tex]

For cylinder B:

[tex]V_B=2h*(2r)^2*\pi[/tex]

Simplify the equation for volumen of cylinder B:

[tex]\begin{gathered} V_B=2h*4r^2*\pi \\ V_B=8*(h*r^2*\pi) \end{gathered}[/tex]

in the equation for the volume of cylinder A you have the value of h*r^2*π:

[tex]\begin{gathered} V_B=8*(10\pi cm^3) \\ V_B=80\pi cm^3 \end{gathered}[/tex]Then, the volume of cylinder B is 80π cubic centimeters.

Since the collection contains quarters and dimes, we are going to compare the whole collection to the quarters (7),

We can represent a ratio, like:

15:7 or 15/7

The linear equation parallel to y= 3x + 7 is:

y = 3x - 2

How to find the linear equation?

A general linear equation is of the form:

y = m*x + b

Where m is the slope and b is the y-intercept.

Two lines are parallel only if the lines have the same slope and different y-intercepts.

So a line parallel to y = 3x + 7 will be of the form:

y = 3x + c

To find the value of c we use the point (3, 7) which must belong to the line, replacing the values in the linear equation:

7 = 3*3 + c

7 = 9 + c

7 - 9 = c

-2 = c

The linear equation is y = 3x - 2

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Speed formula:

[tex]s=\frac{d}{t}[/tex]

d is the distance

t is the time

As you need to find a time having the distance and speed, solve the equation above for t:

[tex]\begin{gathered} t\cdot s=d \\ t=\frac{d}{s} \end{gathered}[/tex]

Use the given data to find the time:

[tex]\begin{gathered} t=\frac{278mi}{38mi/h} \\ \\ t=7.31h \\ \\ t=7h \end{gathered}[/tex]Then, it would take you 7 hours to drive 278 mi at 38mi/h

To answer this question we have to find (among the options) the amount that represents half the amount of water used.

Since the ratio of water to vinegar is 2:1, half of the amount of water will be used of vinegar.

In this case we have to find the answer that represents half a gallon.

That answer is 2 quarts. 2 guarts are 0.5 gallons, it means they are half the amount of water used.

It means that the answer is b. 2 quarts.

ANSWER

The cofunction of tan 36 degrees is cot 54 degrees

STEP-BY-STEP EXPLANATION

Given information

[tex]\text{tan 26}\degree[/tex]

Co function of tan can be written below as

[tex]\begin{gathered} \tan \text{ }(A)\text{ = cot (B)} \\ \text{if, A + B = 90} \end{gathered}[/tex][tex]\begin{gathered} \text{tan 36 = cot (90 - 36)} \\ \tan \text{ 36 = cot 54} \end{gathered}[/tex]

Therefore,

tan 36 = 0.7265

cot 54 = 0.7265

Hence, the cofunction of tan 36 is cot 54

To find the coordinates of the midpoint

We will use the formula;

[tex](x_m,y_m)=(\frac{x_1+x_2}{2},\text{ }\frac{y_1+y_2}{2})[/tex]

x₁ = -4 y₁=7 x₂ = 0 y₂=-3

substituting into the formula

Xm = x₁+x₂ /2

=-4+0 /2

=-2

Ym= y₁+ y₂ /2

=7-3 /2

=4/2

=2

The coordinates of the midpoint m are (-2, 2)

ANSWER

EXPLANATION

a) To find the time it will take before he has $2400, we have to apply the formula for monthly compounded amount:

[tex]undefined[/tex]

Answer:

(8, 0)

Explanation:

Whenever a point (x,y) is reflected across the line y=x, the transformation rule is given below:

[tex](x,y)\to(y,x)[/tex]

That is, the x-coordinate and y-coordinate change places.

Therefore, the image of the point (0,8) when reflected across the line y=x is:

[tex](8,0)[/tex]

The correct answer is (8,0).

Answer:

P(X<16) = 0.64P(X>12) = 0.64

Step-by-step explanation:

Given a graph of a probability density function, you want the probabilities ...

P(X < 16)P(X > 12)

Probability from PDF

The probability of a given range of values of X is the area under the density curve for those values of x.

P(X < 16)

The triangular area to the left of X=16 has a base of 16 and a height of 0.08. Its area is given by the area formula for a triangle:

A = 1/2bh

A = 1/2(16)(0.08) = 0.64

The probability is P(X<16) = 0.64.

P(X > 12)

The area to the right of X=12 is a trapezoid with parallel "bases" of 0.06 and 0.10. The "height" of the trapezoid is 20-12 = 8. The area is given by the formula ...

A = 1/2(b1 +b2)h

A = 1/2(0.06 +0.10)(8) = 0.64

The probability is P(X>12) = 0.64.

Given:

[tex]\frac{70}{45}[/tex]

Required:

We need to convert the given fraction to a decimal.

Explanation:

Divide the number 70 by 45.

[tex]\frac{70}{45}=1.555...[/tex]

Round off to the nearest hundredth.

[tex]\frac{70}{45}=1.56[/tex]

Final answer:

[tex]\frac{70}{45}=1.56[/tex]

Given:

[tex]\frac{70}{45}[/tex]

Required:

We need to convert the given fraction to a decimal.

Explanation:

Divide the number 70 by 45.

[tex]\frac{70}{45}=1.555...[/tex]

Round off to the nearest hundredth.

[tex]\frac{70}{45}=1.56[/tex]

Final answer:

[tex]\frac{70}{45}=1.56[/tex]

To find the asymptotes, we have to solve the following.

[tex]x^2-4x+3=0[/tex]

We have to find two numbers whose product is 3 and whose sum is 4. Those numbers are 3 and 1.

[tex](x-3)(x-1)=0[/tex]

So, the solutions are x = 3 and x = 1.

Hence, the asymptotes x = 1 and y = 1/2.

The graph below shows the function.

Transformation of a Function

We are given the function:

[tex]y=f(x)=\frac{2}{3}x+8[/tex]

And it's required to find another function g(x) according to the transformation:

g(x) = -f(x) - 2

First, we calculate the negative of f(x):

[tex]-f(x)=-(\frac{2}{3}x+8)=-\frac{2}{3}x-8[/tex]

And now we subtract 2 to find g(x):

[tex]g(x)=-\frac{2}{3}x-8-2=-\frac{2}{3}x-10[/tex]

The equation above is in the slope-intercept form where the slope is m=-2/3 and the y-intercept = -10

Answer:

[tex]g(x)=-\frac{2}{3}x-10[/tex]

7/11% can be written as:

[tex]\begin{gathered} \frac{\frac{7}{11}}{100}=\frac{7}{11}\times\frac{1}{100} \\ \frac{\frac{7}{11}}{100}=\frac{7}{1100} \end{gathered}[/tex]

So 7/11% of a quantity is equal to 7/1100 fraction of a quantity ​

Answer:41/10,000

7/17% = 0.41% = 0.0041

41/10,000

Step-by-step explanation:

A: 90% - 100%

B: 80% - 89%

C: 70% - 79%

D: 60% - 69%

F: 0% - 59%

Using the data provided:

[tex]0.5(\frac{60+90+72}{3})+0.1(68)+0.2(64)+0.2(x)\ge70[/tex]

Where:

x = Score of the final exam in order to get at least a C.

Solve for x:

[tex]\begin{gathered} 37+6.8+12.8+0.2x\ge70 \\ 56.6+0.2x\ge70 \\ 0.2x\ge70-56.6 \\ 0.2x\ge13.4 \\ x\ge\frac{13.4}{0.2} \\ x\ge67 \end{gathered}[/tex]

He needs to score between 67 and 100

[tex]-1\frac{3}{4}[/tex] is located at a point 1 on the number line.

1.1125 is located at point 6 on the number line.

14 / 8 is located at point 8 on the number line.

-0.875 is located at point 3 on the number line.

What are the locations of the numbers?

The numbers are made up of mixed fractions, improper fractions, decimals, positive numbers and negative numbers.

A mixed number is a number that has a whole number, a numerator and a denominator. The numerator that has a smaller value than the denominator. and a proper fraction. . An example of a mixed number is 1 1/4. An improper fraction is a fraction in which the numerator is bigger than the denominator. An example of an improper fraction is 14/8.

A negative number is a number that is smaller in value than 0. Negative numbers would be to the left of zero on number line. An example of a negative number is -1.4. A positive number is a number that is greater in value than 0. Positive numbers are located to the right of 0 on the number line. An example of a positive number is 4.2.

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[tex]-1\frac{3}{4}[/tex] is located at a point 1 on the number line.

1.1125 is located at point 6 on the number line.

14 / 8 is located at point 8 on the number line.

-0.875 is located at point 3 on the number line.

What are the locations of the numbers?

The numbers are made up of mixed fractions, improper fractions, decimals, positive numbers and negative numbers.

A mixed number is a number that has a whole number, a numerator and a denominator. The numerator that has a smaller value than the denominator. and a proper fraction. . An example of a mixed number is 1 1/4. An improper fraction is a fraction in which the numerator is bigger than the denominator. An example of an improper fraction is 14/8.

A negative number is a number that is smaller in value than 0. Negative numbers would be to the left of zero on number line. An example of a negative number is -1.4. A positive number is a number that is greater in value than 0. Positive numbers are located to the right of 0 on the number line. An example of a positive number is 4.2.

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Given

Length of hypotenuse= 25.1 cm

length of BC = 16.4 cm

Find

Value of

[tex]sin\beta[/tex]

Explanation

As , we know

[tex]sin\beta=\frac{opposite}{hypotenuse}[/tex]

now, put values

[tex]sin\beta=\frac{16.4}{25.1}=0.653[/tex]

Final Answer

Value of

[tex]sin\beta=0.653\text{ approx}[/tex]

Step 1

Given;

[tex]\begin{gathered} Head\text{ represent male} \\ Tail\text{ represent female} \end{gathered}[/tex]

The total number of puppies is 4 represented by 4 coins.

Step 2

Find the experimental probability that exactly 3 of the puppies will be female

[tex]\begin{gathered} From\text{ table we find that THTT, TTHT, HTTT and HTTT are the only outcomes that } \\ \text{show exactly 3 females} \\ Remember\text{ tail\lparen t\rparen is for female puppies} \end{gathered}[/tex]

Therefore, the total number of samples/coin tosses=20

The formula for probability is;

[tex]Pr\left(event\right)=\frac{Numberofrequiredevent}{Total\text{ number of events}}[/tex]

Total number of events =the total number of samples/coin tosses=20

Number of required events= outcomes with 3 T's from the tab;e=4

Hence.

[tex]=\frac{4}{20}=0.2=0.2\times100=20\text{\%}[/tex]

Answer;

[tex]\frac{4}{20}=0.20=20\text{\%}[/tex]

To get the better deal of the two, we need to find the cost per area of pizza for each deal and compare.

Deal 1: 2 medium 14'' (round) pizza for $14 total

The area of a circle is calculated as

[tex]A=\pi r^2[/tex]

where r is the radius.

The area of the pizza is calculated to be:

[tex]\begin{gathered} r=14 \\ \therefore \\ A_1=\pi\times14^2=196\pi \end{gathered}[/tex]

Hence, the total area for the two pizzas will be:

[tex]\Rightarrow196\pi\times2=392\pi[/tex]

The cost per square inch of pizza is, therefore, calculated to be:

[tex]\Rightarrow\frac{14}{392\pi}=0.011[/tex]

The pizza costs $0.011 per square inch.

Deal 2: 1 large 20'' (round) pizza for $13 total

The area of the pizza is calculated to be:

[tex]\begin{gathered} r=20 \\ \therefore \\ A_2=\pi\times20^2=400\pi \end{gathered}[/tex]

Hence, the cost per square inch of pizza is calculated to be:

[tex]\Rightarrow\frac{13}{400\pi}=0.010[/tex]

The pizza costs $0.010 per square inch.

CONCLUSION:

The better deal will be the deal with the lesser cost per square inch. As can be seen from the calculation, both deals are about the same price per square inch if approximated. However, without approximation, Deal 2 has a slightly lesser cost per square inch.

Therefore, DEAL 2 IS THE BETTER DEAL.

Given:

lim (2x² - x + 3)/(3x² + 5)

We are to

Event A:

The event A occurs when an odd number is rolled in the second roll and in the third roll. We can see in the table that the outcomes that correspond with this event are:

OOO

EOO

Now to calculate the probability, we need to divide the number of favorable outcomes by the number of total outcomes. There are 8 possible outcomes, and the favorable outcomes for event A are 2. Thus:

[tex]P(A)=\frac{2}{8}=\frac{1}{4}[/tex]

Event B:

In event B we want the last roll to be even. Then, the outcomes corresponding to this event are:

OOE

EEE

EOE

OEE

The number of favorable outcomes is 4, the total outcome is 4:

[tex]P(B)=\frac{4}{8}=\frac{1}{2}[/tex]

Event C:

Here, we are looking for outcomes with an even number in the second or last roll (or both). Thus the outcomes that satisfy this are:

OOE

EEE

OEO

EEO

EOE

OEE

The number of favorable outcomes is 6, and the number of total outcomes is 8:

[tex]P(C)=\frac{6}{8}=\frac{3}{4}[/tex]

ANSWER:

STEP-BY-STEP EXPLANATION:

a.

We know that the net profit is equal to the incomes minus the expenses, therefore, the final equation would be:

[tex]\begin{gathered} \text{profit = income - expense} \\ \text{replacing} \\ p=-0.3t^2+8t+198-(-0.2t^2+2t+131) \\ p=-0.3t^2+8t+198+0.2t^2-2t-131 \\ p=-0.1t^2+6t+67 \end{gathered}[/tex]

b. t is the number of the years after 2010. Therefore, for the year 2016, x is equal to 6 (2016 - 2010), we replace:

[tex]undefined[/tex]

AS shown in the figure:

The measure of arc RT = 27

The measure of arc FN = 105

The measure of angle FUN will be as follows:

[tex]m\angle\text{FUN}=\frac{1}{2}(105+27)=\frac{1}{2}\cdot132=66[/tex]

So, the answer is option C. 66

The total number of equally-sized tiles on a circular floor.

Here, we are covering the region or the total space occupied by all the tiles on the floor.

Hence, the area is calculated.

The midpoint m of z₁ = (9+7i) and Z₂ = (-7+7i) is 1 + 7i .

Given complex numbers:

[tex]z_{1}[/tex] = (9 + 7i) and [tex]z_{2}[/tex] = (-7 + 7i)

compare these numbers with a1+ib1 and a2+ib2, we get

a1 = 9, a2 = -7 , b1 = 7 and b2 = 7.

Mid point of complex numbers = a1 + a2 /2 + (b1 + b2 /2)i

= (9 + (-7)/2 + (7 + 7 /2)i

= 2/2 + 14/2 i

Mid point m = 1 + 7i

Therefore the midpoint m of z₁ = (9+7i) and Z₂ = (-7+7i) is 1 + 7i

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