FUNCTIONS: In the space provided, type the answer in descending order as it applies without any spaces between the letters, numbers, or symbols.
Type the composition (fog)(x) of the given functions:
f(x) = x^2 + 2x − 6 and g(x) = x + 5.

Answer:

3

Step-by-step explanation:

The average rate of change of f(x) in the closed interval [ a, b ] is

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

Here [ a, b ] = [ 1, 2 ]

f(b) = f(2) = 2² - 3 = 4 - 3 = 1

f(a) = f(1) = 1² - 3 = 1 - 3 = - 2, thus

average rate of change = [tex]\frac{1-(-2)}{2-1}[/tex] = 3

Answer:

3

Step-by-step explanation:

Average rate of change

= [f(2)-f(1)] ÷ (2-1)

= [(2²-3) - (1²-3)] ÷ 1

= (4-3) - (1-3)

= 1 - (-2)

= 1 + 2

= 3

Answer:

8

Step-by-step explanation:

The distance between the aquarium and the monkey habitat is 325 feet.

To find the distance between the aquarium and the monkey habitat, we can use the Pythagorean theorem because the aquarium and the monkey habitat form a right triangle with the gift shop.

Let's denote the distance between the aquarium and the gift shop as  a (36 feet) and the distance between the monkey habitat and the gift shop as b (323 feet). The distance between the aquarium and the monkey habitat, which we'll call c, is the hypotenuse of the right triangle.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse  c  is equal to the sum of the squares of the lengths of the other two sides a and b.

[tex]\[ c^2 = a^2 + b^2 \][/tex]

Substitute the given values:

[tex]\[ c^2 = (36)^2 + (323)^2 \][/tex]

[tex]\[ c^2 = 1296 + 104329 \][/tex]

[tex]\[ c^2 = 105625 \][/tex]

Take the square root of both sides to solve for  c:

[tex]\[ c = \sqrt{105625} \][/tex]

[tex]\[ c = 325 \][/tex]

Answer:

With side lengths 16 m, 21 m and 39 m a triangle cannot be formed.

Step-by-step explanation:

Given:

Sides of the triangle are given.

First side = 16 m

Second Side = 21 m

Third side = 39 m

We need to find whether the side length can form the triangle.

Solution:

Now we know that;

By Triangle In equality property which states that;

"The sum of length of any two side of triangle is greater than the length of the thirds side."

By applying the same in given data we get;

First Side + second side > Third side

[tex]16+21>39\\\\37>39[/tex]

which is false.

The first condition has failed itself.

So we can say that;

With side lengths 16 m, 21 m and 39 m a triangle cannot be formed.

The side lengths 16m, 21m, and 39m do not form a triangle as they do not satisfy the triangle inequality theorem, which states that for any three lengths to form a triangle, the sum of any two lengths must be greater than the third length.

In mathematics, specifically in triangle geometry, the triangle inequality theorem states that for any given three sides, the lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. In other words, if a, b, and c are the lengths of the sides:

a + b > c

a + c > b

b + c > a

Our given side lengths are 16m, 21m, and 39m. Let's check if they satisfy the triangle inequality:

16m + 21m > 39m (Is 37m > 39m? No, 37m is less than 39m, thus the inequality is not satisfied)

Therefore, the given side lengths do not form a triangle.

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Answer:

26.57 degrees

Step-by-step explanation:

So we need to find the angle of D in your triangle.

Since this is a right triangle we can use trigonometric functions (sin, cos, tan etc)

For this we need to use tangent since wanna do opposite over adjacent.

But since we want the angle we use tan^-1

So plug into your calculator(make sure your calculator is in degrees): tan^-1 (4/8)

We get 26.565 degrees

Answer:

Step-by-step explanation:

Triangle BCD is a right angle triangle.

From the given right angle triangle,

BD represents the hypotenuse of the right angle triangle.

With m∠D as the reference angle,

CD represents the adjacent side of the right angle triangle.

BC represents the opposite side of the right angle triangle.

To determine m∠D, we would apply

the tangent trigonometric ratio.

Tan θ = opposite side/adjacent side. Therefore,

Tan D = 4/8 = 0.5

m∠D = Tan^-1(0.5)

m∠D = 26.6° to the nearest tenth.

Answer:

1.778 times more or 16/9 times more

Step-by-step explanation:

Given:

- Mirror 1: D_1 = 8''

- Mirror 2: D_2 = 6"

Find:

Compare the light gathering power of an 8" primary mirror with a 6" primary mirror. The 8" mirror has how much light gathering power?

Solution:

- The light gathering power of a mirror (LGP) is proportional to the Area of the objects:

                                           LGP ∝ A

- Whereas, Area is proportional to the squared of the diameter i.e an area of a circle:

                                           A ∝ D^2

- Hence,                              LGP ∝ D^2

- Now compare the two diameters given:

                                           LGP_1 ∝ (D_1)^2

                                           LGP ∝ (D_2)^2

- Take a ratio of both:

                           LGP_1/LGP_2 ∝ (D_1)^2 / (D_2)^2

- Plug in the values:

                               LGP_1/LGP_2 ∝ (8)^2 / (6)^2

- Compute:             LGP_1/LGP_2 ∝ 16/9 ≅ 1.778 times more

Answer:

The answer to your question is  A = 74 ft³, B = 222 ft³, C = 222ft³

Step-by-step explanation:

Data

Total volume = 518 ft³

Prism A volume = Prism B volume / 3

Prism B volume = Prism C volume

Process

Write an equation in terms of volume of B

          Volume A + Volume B + Volume C = 518

Substitution

                B/3 + B + B = 518

                (B + 3B + 3B)/3 = 518

                               7B = 3(518)

                               7B = 1554

                                 B = 1554/7

                                 B = 222

Get the volumes

Volume A = 222/3

                 = 74

Volume B = 222

Volume C = 222

Answer:

Therefore the interest = $ 75.

Step-by-step explanation:

Given , Chris shopper received a loan $1,500 discount to purchase a washer and dryer. The loan was offered at 15%for 120 day.

P = $ 1500

r = 15%

t= [tex]\frac{120}{360} = \frac{1}{3}[/tex] year

[tex]\textrm{intrest} =\frac{Pr t}{100}[/tex]

          [tex]=\$\frac{1500\times 15 \times \frac{1}{3} }{100}[/tex]

         [tex]=\$ 75[/tex]

Therefore the interest = $ 75.

Answer:

Option c.

Step-by-step explanation:

We have the set of real numbers greater than 8 and B the set of real numbers greater than 5 here:

A: x+2>10 ⇒ x>8

B: 2x > 10 ⇒ x>5

8∉A ; 8∈B

Note: We can see (graph) that the intervals are open, so the corresponding points do not belong to their respective sets.

Answer:

Option C.300

Step-by-step explanation:

we know that

To find out how many times greater is the land area of the United States than that of Norway, divide the area of the United States by the area of the Norway

so

[tex]\frac{9\times10^6}{3\times10^4}[/tex]

Remember that

To divide exponents (or powers) with the same base, subtract the exponents

so

[tex]\frac{9\times10^6}{3\times10^4}=(\frac{9}{3})\times(10^{6-4} )=3\times10^2=300[/tex]

For a semi-circle with diameter 10, the circumference is 5π, perimeter is 5π + 10, and area is 25/2π. For a circle with radius 6 and a 90-degree cut, remaining circumference is 9π, perimeter is 12π, and remaining area is 27π.

Semi-circle with Diameter 10:

a. Circumference:

The circumference of a circle is given by the formula C = π × d, where d is the diameter. For a semi-circle, it's half of the circumference of a full circle. Thus, C(semi-circle) = π × 10/2 = 5π.

b. Perimeter:

The perimeter of a shape is the sum of all its sides. For a semi-circle, it includes the curved boundary and the diameter. Therefore, P(semi-circle) = 5π + 10.

c. Area:

The area of a semi-circle is given by A(semi-circle) = 1/2 × π × r^2, where r is the radius. Substituting the given diameter, A(semi-circle) = 1/2 × π × (10/2)^2 = 25/2 × π.

Circle with Radius 6 and a 90-Degree Cut:

a. Circumference:

For the full circle, C(circle) = 2π × r = 2 × π × 6 = 12π. Since a 90-degree cut removes a quarter of the circle, the remaining circumference is C(remaining) = 3/4 × 12π = 9π.

b. Perimeter:

The perimeter includes the cut portion, so P(circle) = 12π.

c. Area:

The area of the circle is A(circle) = π × r^2 = π × 6^2 = 36π. With the cut, A(remaining) = 3/4 × 36π = 27π.

Answer:

It should be option B

Step-by-step explanation:

To find the number of students the North and South campuses had before the merger, set up an equation and solve for the unknowns. Use the given percentages and total number of students to determine the number of music majors at each campus.

To find the number of students the North and South campuses had before the merger, we can set up two equations based on the given information. Let's assume the number of students at the North campus is N and the number of students at the South campus is S.

From the information provided, we know that 30% of the students at the North campus are music majors, so the number of music majors at the North campus is 0.3N. Similarly, 80% of the students at the South campus are music majors, so the number of music majors at the South campus is 0.8S.

Since the campuses are merged into the East campus, which has 1000 students and 45% of them are music majors, we can set up the equation 0.45(1000) = 0.3N + 0.8S to represent the total number of music majors at the East campus. From this equation, we can solve for N and S to find the number of students at the North and South campuses before the merger.

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Answer:

Juice $0.6

Bagel $1.5

Coffee $0.3

Step-by-step explanation:

The original price is $2.40 for an orange juice (x), raisin bagel (y) and a cup of coffee(z).

The prices of the orange juice will increase by 50%:

[tex]=x+0.5x=1.5x[/tex]

The price of bagels will increase by 20%:

[tex]=y+0.2x=1.2x[/tex]

The total price of everything is $3.00.

The orange juice will cost twice as much as coffee:

[tex]x=2z[/tex]

We have the following equations:

[tex]x+y+z=2.40[/tex]

[tex]1.5x+1.2y+z=3[/tex]

[tex]x=2z[/tex]

We have 3 equations and 3 unknowns:

Solve through substitution of x=2z into both equations:

[tex]1.5x+y=2.40[/tex]

[tex]2x+1.2y=3[/tex]

Therefore before increase. Use equation before increase to solve z and not x=2z because this is after the increase.

[tex]x=0.6[/tex]

[tex]y=1.5[/tex]

[tex]z=0.3[/tex]

The original prices were $1.00 for orange juice, $0.60 for a raisin bagel, and $0.80 for coffee.

Let's denote the price of orange juice as O, the price of a raisin bagel as B, and the price of coffee as C.

Initially, we know that O + B + C = $2.40.

After the price increase, we know that (1.5O) + (1.2B) + C = $3.00 and also that 1.5O = 2C.

From the equation 1.5O = 2C, we deduce that O = 1.33C. We substitute this in the first equation to get the initial prices:

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Answer:

Brian Peters worked for total of 45 hours.

Step-by-step explanation:

Let the total number of hours worked be 'x'.

Now Given:

Hours spent on other projects = 18 hours.

Also Given:

60% of a week's time working on drawings for a new apartment building.

Hours spent on new apartment building = [tex]60\%\times x = \frac{60}{100}x=0.6x[/tex]

We need to find the total hours worked.

Solution:

Now we can say that;

total number of hours worked is equal to sum of Hours spent on new apartment building and Hours spent on other projects.

framing in equation form we get;

[tex]x=0.6x+18[/tex]

Combining like terms we get;

[tex]x-0.6x=18\\\\x(1-0.6)=18\\\\0.4x=18[/tex]

Now Dividing both side by 0.4 we get;

[tex]\frac{0.4x}{0.4}=\frac{18}{0.4}\\\\x=45\ hrs[/tex]

Hence Brian Peters worked for total of 45 hours.

In the question, architect Brian Peters spent 60% of his total work time on a new apartment building and 40% on other projects. If those 40% equates to 18 hours, we can calculate his total work week as 45 hours by dividing 18 by 0.4.

The problem can be solved by identifying that architect Brian Peters spent 65% of his time working on other projects than the new apartment building. This implies that he spent 40% of his time on the apartment building project. Understanding that a week consists of 168 hours, we could conduct a calculation to determine his total work hours. If we take that 60% was spent on the apartment building and 40% was spent on other projects, then those 18 hours represent 40% of his total work time. Therefore, to find his total work time, we can divide 18 (hours spent on other projects) by 40%, or 0.4.

So, the calculation will be: 18/0.4 = 45 hours. This shows that Brian Peters worked a total of 45 hours within the week for all his projects. These 45 hours involve the 18 hours he spent on other projects and the remaining time he dedicated to the new apartment building drawings.

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Number of tires that were sold is 15 and number of rims that were sold is 5. Total sales for tires is $2250 and total sales for rims is $1000

Step-by-step explanation:

Given details are selling price of tires = $150, selling price of rims = $200 and total dales = $3250

Form equation out of the given data. Let the number of rims that were sold be x. Then number of tires that were sold is 3x.

⇒ x × 200 + 3x × 150 = 3250

⇒ 200x + 450x = 3250

⇒ 650x = 3250

⇒ x = 5

Find number of tires and rims that were sold.

⇒ Number of rims that were sold = x = 5

⇒ Number of tires that were sold = 3x = 15

Find the total sales for each.

⇒ Total sales for rims = 5 × 200 = $1000

⇒ Total sales for tires = 15 × 150 = $2250

5 rims and 15 tires were sold with total sales of $1,000 for rims and $2,250 for tires.

Let x be the number of rims sold and 3x be the number of tires sold, since three times as many tires were sold as rims.

Now, we can create equations based on the given information.

The total sales for rims and tires are $3,250.

x = $3,250 / $650

3x = 3 * 5 = 15 (the number of tires sold)

Total sales for rims were $200 * 5 = $1,000, and total sales for tires were $150 * 15 = $2,250.

Answer: it will take 9 miles for both costs to be the same.

Step-by-step explanation:

Let x represent the number of miles it will take for both companies to charge the same amount.

Metro taxi charges $3.00 for the first mile and $2.50 for each additional mile. This means that the total charge for x miles would be

3 + 2.5(x - 1) = 3 + 2.5x - 2.5

= 2.5x + 0.5

City taxi charges $5.00 for the first mile and $2.25 for each additional miles. This means that the total charge for x miles would be

5 + 2.25(x - 1) = 5 + 2.25x - 2.25

= 2.25x + 2.75

For the costs to be the same, then

2.5x + 0.5 = 2.25x + 2.75

2.5x - 2.25x = 2.75 - 0.5

0.25x = 2.25

x = 2.25/0.25

x = 9

Answer:

The violent-crime rate in a certain state of the country in that year was 1,496

Would this be an outlier?

C. Yes, because it is greater than the upper fence.

(d) Do you believe that the distribution of violent-crime rates is skewed or symmetric?

C. The distribution of violent-crime rates is skewed right.

Step-by-step explanation:

The violent-crime rate in a certain state of the country in that year was 1,496

The lower fence is 272.8 - 1.5 x 255.5 = -110.45 crimes per​ 100,000 population.

The upper fence is 528.3 + 1.5 x 255.5 = 911.55 crimes per​ 100,000 population.

Since violent-crime rate in this certain state of the country in that year is greater than the upper fence (1496>911.55), then it is an outlier.

(d) Do you believe that the distribution of violent-crime rates is skewed or symmetric?

The distribution of violent-crime rates is not symmetric, as there are extreme values in the​ tail, which tend to pull the mean in the direction of the tail to have data are either skewed left or skewed​ right, in this case it is skewed​ right, as there are large observations in the right tail that tend to increase the value of the​ mean, while having little effect on the median.

Answer:

-4

Step-by-step explanation:

The average rate of change of a function f(x) between two points x = a and x=b is given by,

[tex]f(x)=\frac{f(b)-f(a)}{b-a}[/tex]

This can be understood with a simpler example, the straight line.

In this case, the rate of change of the 'function' is given by the slope,

[tex]m=\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}[/tex];                

[tex](x_{1},f(x_{1})),(x_{2},f(x_{2}))[/tex] being two points on the straight line.

So, for the given problem,

a = -2

b = 2

Hence, average rate of change of [tex]f(x)=\frac{f(2)-f(-2)}{2-(-2)} =\frac{9-25}{2+2} =\frac{-16}{4} =-4[/tex]

Answer:

The slope of the line is:    [tex]$ \frac{\textbf{7}}{\textbf{3}} $[/tex]

Step-by-step explanation:

The product of the slope of two perpendicular slopes = - 1.

Given the slope of one of the perpendicular lines, say, [tex]$ m_{1} $[/tex] = [tex]$ -\frac{3}{7} $[/tex].

We have to determine the slope of the line perpendicular to the first line, We call this slope - [tex]$ m_{2} $[/tex].

We know that the product of the slopes [tex]$ m_{1}. m_{2} = - 1 $[/tex]

[tex]$ \implies -\frac{3}{7} . m_2 = - 1 $[/tex]

[tex]$ \implis m_{2} = - 1 \times - \frac{7}{3} $[/tex]

[tex]$ \therefore m_{2} = \frac{\textbf{7}}{\textbf{3}} $[/tex]

Hence, the answer.

Answer: a) 0.22 b) 0.1363

Step-by-step explanation:

People who contract disease I are= 10%

People who contract disease II are= 15%

People who contract both diseases are= 3%

a)

People who contract has at least one disease needs to be found out so it is given as

P(D1 or D2)= P(D1) + P(D2) - P(D1 and D2)

P(D1 or D2)= 10% + 15% - 3%

P(D1 or D2)= 22%

P(D1 or D2)= 0.22

b)

Conditional probability that randomly selected person will get get diseases given she has contracted at least one disease is gven as

Probability= P(D1 and D2) / P(D1 or D2)

Probability= [tex]\frac{0.03}{0.22}[/tex]

Probability= 13.63%

Probability= 0.1363

Answer:

The answer is an (IRA)

Step-by-step explanation:

The IRA means "Individual retirement account"

Answer:

[tex]R = \dfrac{16}{5}t + \dfrac{1}{5}[/tex]

where R is the R-value and t is the thickness in inches.

Step-by-step explanation:

We are given the following in the question:

The  R-value of fiberglass insulation and f its thickness have a linear relation.

Let R be the R-value and t be the thickness. Then, the equation can be written as:

[tex]R = at + b[/tex]

where a and b are constants.

When t = 3.5, R = 11

[tex]11 = 3.5a + b[/tex]

When t = 6, R = 19

[tex]19 = 6a + b[/tex]

Solving the two equation, using the elimination method, we have,

[tex]19-11 = 6a + b-(3.5a + b)\\8 = 2.5a\\\\\Rightarrow a = \dfrac{16}{5}\\\\11 = 6(\dfrac{16}{5}) + b\\\\\Rightarrow b = \dfrac{1}{5}[/tex]

Thus, the linear relationship is given by:

[tex]R = \dfrac{16}{5}t + \dfrac{1}{5}[/tex]

where R is the R-value and t is the thickness in inches.

To find a linear equation (R = mt + b) that represents the R-value of fiberglass insulation as a function of its thickness, we calculate a slope (m) of 3.2 using two known measurements. We then solve for a y-intercept (b), which is -0.5. Therefore, the equation is R = 3.2t - 0.5.

In order to derive the equation that gives the R-value as a function of the thickness, we need to use the concept of linear equations, specifically slope-intercept form (y = mx + b). First, we need to find the slope (m). Since we know two points on the line (3 ½ , 11) and (6 , 19), we calculate the slope as: m = (19-11) / (6 - 3½) = 8 / 2½ = 3.2. The y-intercept (b) is the R-value when the thickness (t) is 0. To find it, we use one of our points and the slope in the equation y = mx + b, substituting to solve for b: 11 = 3.2 * 3½ + b, hence, b = -0.5. The equation is therefore: R = 3.2t - 0.5.

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Answer:

[tex]W=(x-6)\ m[/tex]

Step-by-step explanation:

we know that

The area of rectangle is equal to

[tex]A=LW[/tex]

we have

[tex]A=(x^{2} -11x+30)\ m^2[/tex]

[tex]L=(x-5)\ m[/tex]

substitute the given values in the formula of area

[tex](x^{2} -11x+30)=(x-5)W[/tex]

Remember that

[tex]x^{2} -11x+30=(x-5)(x-6)[/tex] ----> by completing the square

substitute

[tex](x-5)(x-6)=(x-5)W[/tex]

Simplify

[tex]W=(x-6)\ m[/tex]

Answer: Ollie earned $9 per hour.

Step-by-step explanation:

Let x represent the amount that Ollie earned per hour.

Stan and Ollie each earned the same amount last week Stan work 20 hours and earned $13.50 per hour. This means that the total amount that Stan earned would be

20 × 13.5 = $270

Ollie worked 30 hours. This means that the total amount that Ollie earned would be

30 × x = $30x

Since they earned the same amount, then

30x = 270

x = 270/30 = $9 per hour

Answer:

$9 per hour

Step-by-step explanation:

The difference between terms is the previous term divided by 4:

48/4 = 12

12/4 =3

3/4 = 0.75

Find the next two terms:

0.75/4 = 0.1875

0.1875/4 = 0.046875

Now you have 48 - 12 + 3 - 0.75 + 0.1875 - 0.046875

Answer: 38.390625

Answer:

Every year they are losing 600 people a year.

Step-by-step explanation:

You take 50,000/100= 500 x 1.2=600

Answer:

The value of h is 42.956 approximately.

Step-by-step explanation:

Consider the provided formula [tex]h=\dfrac{0.00252 d^{2.27}}{e}.[/tex]

Here d is the distance of the driver from the letters and e is the height of the​ driver's eye above the pavement. All of the distances are in meters.

We need to find the value of h where the value of d = 92.4 m, e = 1.7 m.

Substitute d = 92.4 m, e = 1.7 m in above formula and solve for h.

[tex]h=\dfrac{0.00252\left(92.4\right)^{2.27}}{1.7}[/tex]

[tex]h\approx\dfrac{0.00252\left(28978.4648\right)}{1.7}[/tex]

[tex]h\approx\dfrac{73.0257}{1.7}[/tex]

[tex]h\approx42.956[/tex]

Hence, the value of h is 42.956 approximately.

Final answer:

Using the formula provided, the optimal height of the letters for a message to be seen by a driver is approximately 30.447 meters, when the driver is 92.4 meters away and their eye height is 1.7 meters above the pavement.

Explanation:

To find the optimal height h of the letters for a message printed on pavement, where d is the distance to the driver and e is the height of the driver's eye above the pavement, we use the given formula: h = [tex]\(\frac{0.00252 d^{2.27}}{e}\)[/tex]

Substituting the provided values d = 92.4 m and e = 1.7 m into the formula, we calculate:

h = [tex]\(\frac{0.00252 \times 92.4^{2.27}}{1.7}\)[/tex]

First, raise 92.4 to the power of 2.27:

92.42.27 ≈ 20546.45

Then, multiply this result by 0.00252:

0.00252 × 20546.45 ≈ 51.76

Finally, divide by e, the driver's eye height (1.7 m):

h ≈ [tex]\(\frac{51.76}{1.7}\)[/tex] ≈ 30.447 m

Therefore, the optimal letter height h is approximately 30.447 meters.