Shannon manages a small zoo and she has been analyzing the attendance data. Shannon finds that the number of visitors increases exponentially as the temperature increases, and this situation is represented by the function f(x) = 3x. Shannon also finds a linear equation that models the number of people who leave the park early depending on the change in temperature, and it is represented by g(x) = −x + 4. The graph of the two functions is below. Find the solution to the two functions and explain what the solution represents.

Graph increasing exponentially from the left, crosses the y axis at one and goes to infinity and intersects a linear graph that decreases and crosses the y axis at four and the x axis at four.

The Pythagorean theorem states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.

Therefore, for the given values of 3, 4, and 5, these lengths can indeed form a right triangle. Specifically, this triangle is a 3-4-5 triangle, which is a special type of right triangle whose sides have integral lengths and whose angles are all multiples of 30°.

Mathematically, this can be expressed as a^2 + b^2 = c^2, where a and b are the lengths of the two legs of the triangle and c is the length of the hypotenuse. The theorem is a fundamental result in geometry and has wide-ranging applications in mathematics, physics, engineering, and other fields.

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The value of the given indefinite integral is found to be: y = -cos (9x² + 8) + c.

The given integral is;

y = ∫18x sin (9x² + 8) dx

Put u = 9x² + 8

Then, on differentiation

18x dx = du

substitute the values:

y = ∫ sin (u) du.

y = -cos u

Put u = 9x² + 8

y = -cos (9x² + 8) + c

c is the constant of integration.

Thus, the value of the given indefinite integral is found to be: y = -cos (9x² + 8) + c.

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Answer: 1, 2, and 4.

Step-by-step explanation: Each must be wider than the angle than a "L" has.

The best numerical descriptive measure for comparing average store sales by region would be the mean or average.

This measure would provide an accurate representation of the overall store sales across the entire region and would allow the manager to easily identify which region has the lowest store sales.

Additionally, the median and mode could also be used as descriptive measures to compare the store sales by region.

For example, if there is a single store with particularly high sales, the mean will be heavily influenced by this one store. The median and mode would be more likely to provide a better representation of the overall sales across the region, as they are not as heavily influenced by extreme values.

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The relation between total salary and saving can be represent by the equation, Y =  X + 292

An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.

For example, 3x+2y=0.

Types of equation

1. Linear Equation

2. Quadratic Equation

3. Cubic Equation

Given that,

Catniss is saving money to take a trip to the Capitol with her sister

Every week,

After taking out the $292, which she needs for her weekly expenses

And that amount she sets aside the same portion of her paycheck

After 10 weeks ,

Catniss has $405

Suppose, she is saving X amount every week,

And her total salary is, Y

Total salary = saving + weekly expenses

    Y            =  X + 292

So, she saved money after 10 week,

⇒ 10X

Given saving after weeks, $405

⇒ 10X = 405

⇒  X = 40.5

Y =  X + 292

   =  40.5 + 405

   =   $445.5

Hence, the equation is Y =  X + 292

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The set of solutions for the linear system is x1 = 6, x2 = 3, and x3 = -4.

The linear system is given as follows:

x1 + x2 - x3 = 6

2x1 + x2 + 2x3 = 8

3x1 + 2x2 + 3x3 = 14

To solve this system, we will use the Gauss-Jordan elimination method. First, we will subtract twice the first equation from the second equation and three times the first equation from the third equation.

2x1 + x2 + 2x3 = 8

- 2(x1 + x2 - x3 = 6)

2x1 + x2 + 2x3 = 8

3x1 + 2x2 + 3x3 = 14

- 3(x1 + x2 - x3 = 6)

0x1 + 3x2 + 5x3 = 6

Next, we will subtract three times the second equation from the third equation.

3x1 + 2x2 + 3x3 = 14

- 3(0x1 + 3x2 + 5x3 = 6)

3x1 + 2x2 - 2x3 = 8

Finally, we will divide the first equation by 1, the second equation by 3, and the third equation by -2.

x1 + x2 - x3 = 6

/ 1

x1 + (1/3)x2 + (5/2)x3 = 3

(-1/2)x1 + x2 + x3 = -4

The set of solutions for the linear system is x1 = 6, x2 = 3, and x3 = -4.

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The solution is Option C.

The measure of TR of triangle STR is TR = 30

If two triangles' corresponding angles are congruent and their corresponding sides are proportional, they are said to be similar triangles. In other words, similar triangles have the same shape but may or may not be the same size. The triangles are congruent if their corresponding sides are also of identical length.

Corresponding sides of similar triangles are in the same ratio. The ratio of area of similar triangles is the same as the ratio of the square of any pair of their corresponding sides

Given data ,

Let the first triangle be represented as ΔPQR

Let the second triangle be represented as ΔSTR

The measure of PR = 35

The measure of SR = 25

The measure of QR = 42

The measure of QT = x

The measure of TR = 42 - x

Now , PQ is similar to ST

Now , the triangles PQR and STR are similar triangles

And , corresponding sides of similar triangles are in the same ratio

Substituting the values in the equation , we get

SR / PR = TR / QR

25 / 35 = ( 42 - x ) / 42

On simplifying the equation , we get

Multiply by 42 on both sides of the equation , we get

( 42 - x ) = ( 25 x 42 ) / 35

On further simplification , we get

42 - x = 30

Adding x on both sides of the equation , we get

x + 30 = 42

Subtracting 30 on both sides of the equation , we get

x = 12

Now , the measure of TR = 42 - x

Substitute the value of x in the equation , we get

The measure of TR = 42 - 12 = 30

Hence , the measure of TR of triangle is 30

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Sadie's total weekly sales in the last week were $625.

A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.

Given, Sadie earns a 20 percent commission on sales at a hair salon.

Let, 'x' be her total weekly sales.

Therefore, $125 is 20% of 'x' which can be numerically expressed as,

(20/100)×x = 125.

0.2x = 125.

x = 125/0.2.

x = 625.

So, her weekly sale was $625.

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The only items she can add to the box to make it less than 12 pounds are;  either the package of socks or the book

The list of items and their weight that she has already packed are;

Boots = 5.2 pounds

Jeans = 1.25 pounds

Sweater = 2.05 pounds

Running Shoes = 2.25 pounds

We are told that the package must weigh less than 12 pounds. Thus, let the weight of the additional item be x and as such we have the inequality;

5.2 + 1.25 + 2.05 + 2.25 + x < 12

10.75 + x < 12

Subtract 10.75 from both sides to get;

x < 1.25

The only item that is less than 1.25 pounds that she can carry is either the package of socks or the book

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Answer:

Step-by-step explanation:

The value of the given expression will be 4.

The expression that represents the combined inventory of these two stores is given as follows:

A. 7g²/2 - 4g/5 + 15/4.

The combined inventory is obtained adding the inventories for each store, combining the like terms.

The addition of the like terms for g² is given as follows:

1/2 + 3 = 0.5 + 3 = 3.5 = 7/2g².

The term with g is given as follows:

-4g/5.

The constant like terms are combined as follows:

7/2 + 1/4 = 14/4 + 1/4 = 15/4.

Hence the simplified expression is given as follows:

7g²/2 - 4g/5 + 15/4.

Meaning that the correct option is given by option A.

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the random variable x is the number of occurrences of an event over an interval of 10 minutes. it can be assumed the probability of an occurrence is the same in any two time periods of an equal length. it is known that the mean number of occurrences in 10 minutes is 5.3.

A Poisson process is a statistical model used to describe the number of occurrences of an event over a fixed interval of time or space. It has the following properties:

The events are independent, meaning that the occurrence of one event does not affect the probability of another event happening.

The average rate of occurrences is constant over time and is equal to the rate parameter of the Poisson distribution.

The probability of having n events in a fixed interval of time is given by the Poisson probability formula:

P(n) = (λ^n * e^-λ) / n!

where λ is the rate parameter and n! is the factorial of n.

Given the mean number of occurrences in 10 minutes is 5.3, it can be assumed that the rate parameter is 0.53 occurrences per minute. This means that, on average, 0.53 events are expected to occur in each minute. Since the Poisson process is memoryless, this rate is constant over time and the mean number of occurrences in any 10-minute interval will be 5.3.

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The Venn diagram shown to Region II: Items in A and B, but not C are (A ∪ B)' ∪ A'

We have been given that the diagram we need to find the regions represent the set.

A Venn diagram is an illustration that uses circles to show the relationships among things or finite groups of things. Circles that overlap have a commonality while circles that do not overlap do not share those traits. Venn diagrams help to visually represent the similarities and differences between two concepts.

The Venn diagram, item represents the regions four, five, six and seven.

Therefore, it can be written us for Union Fight,...., we concluded that the regions represent see R. C.

This is equal to for Union Fighting Union Six, Seven.

We have, region represent (A ∪ B)' ∪ A'

Therefore, the Venn diagram shown to regions represent

(A ∪ B)' ∪ A' is Region II

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The solution to the equations -5m + 6n = 25 and -7m + 8n = 33 is (1, 5)

An equation is an expression that shows the relationship between numbers and variables using mathematical expressions. Equations are classified based on degree as linear, quadratic, cubic

a) Given the equations:

-5m + 6n = 25    (1)

And:

-7m + 8n = 33     (2)

From both equations simultaneously:

m = 1, n = 5

The solution is (1, 5)

b) Given the equations:

8x - 3y = -16    (1)

And:

50 = -9x - 2y     (2)

From both equations simultaneously:

x = -4.23, y = -5.95

The solution is (-4.23, -5.95)

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Answer:

[tex]43.78^\circ[/tex]

Step-by-step explanation:

Let θ  be the angle that the base of the leaning tower makes with the tower i.e. the angle of elevation

Since this is a right triangle we have the relationship

[tex]\tan\theta = \dfrac{height}{base} = \dfrac{46}{48} \approx 0.95833\\\\[/tex]

[tex]\theta = tan^{-1} (0.95833) = 43.78^\circ[/tex]

Answer
[tex]= 43.78^\circ[/tex]

The angle of elevation is given by the trigonometric relation and θ = 73.40°

What are trigonometric relations?

Trigonometry is the study of the relationships between the angles and the lengths of the sides of triangles

The six trigonometric functions are sin , cos , tan , cosec , sec and cot

Let the angle be θ , such that

sin θ = opposite / hypotenuse

cos θ = adjacent / hypotenuse

tan θ = opposite / adjacent

tan θ = sin θ / cos θ

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Given data ,

Let the angle of elevation be represented as E = θ

Now , the measure of the altitude = 46 m

The measure of the hypotenuse = 48 m

And , the angle of elevation is given by the trigonometric relation ,

sin θ = opposite / hypotenuse

On simplifying , we get

sin θ = 46 / 48

sin θ = 0.958333

Taking inverse on both sides , we get

θ = sin ⁻¹ ( 0.958333 )

θ = 73.40°

Hence , the angle of elevation is 73.40°

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The equation of a line that passes through (0.5, 4.25) and (2, 18.5) and has a y-intercept of -0.5 is y = 9.5x - 0.5

The equation of a line is given by:

y = mx + c

where m is the slope of the line and c is the y-intercept.

Example:

The slope of the line y = 2x + 3 is 2.

The slope of a line that passes through (1, 2) and (2, 3) is 1.

We have,

The equation of a line that passes through (0.5, 4.25) and (2, 18.5).

y-intercept =  -5

Now,

The equation of a line is y = mx + c

c = y-intercept

c = -0.5 ______(1)

Now,

(0.5, 4.25) and (2, 18.5)

m = (18.5 - 4.25) / (2 - 0.5)

m = 14.25 / 1.5

m = 9.5 ______(2)

Now,

The equation of a line.

y = mx + c
From (1) and (2).

y = 9.5x - 0.5

Thus,

y = 9.5x - 0.5 is the equation of the line.

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Answer:

Domain

Step-by-step explanation:

The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x).

The Gourmet Grill Co. produces propane and electric grills with varying cost and hours required in 4 manufacturing processes. Aim is to maximize profit by determining the production plan using LP model, feasible region and level curves.

a. The LP model for this problem can be formulated as follows:

Maximize: $320P + $260E (profit from selling propane and electric grills)

Subject to:

2P + E <= 2400 (hours of machine press time available)

4P + 5E <= 6000 (hours of fabrication time available)

2P + 3E <= 3300 (hours of assembly time available)

P + E <= 1500 (hours of testing time available)

P, E >= 0 (non-negativity constraints)

Where P is the number of propane grills and E is the number of electric grills produced.

b. The feasible region for this problem is a polygon with vertices (0, 0), (600, 0), (300, 900), (0, 1200). This region represents the combination of P and E that satisfies all the constraints.

c. To determine the optimal solution using level curves, we can start by plotting the feasible region and then finding the level curve that represents the maximum profit. The level curve would be a straight line with slope equal to $320/$260 (the ratio of price per unit for the two types of grills). The intersection of this line with the feasible region would represent the optimal solution, which would be the combination of P and E that maximizes the profit.

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Answer:

6y=18x+24

Step-by-step explanation:

x come first cuz of pemdas

Answer:

C

Step-by-step explanation:

distance depends on the speed & the time traveled.

The sampling distribution of xM - xB is normally distributed with a mean of 18 mg/dl and a standard deviation of 35.4 mg/dl.

The sampling distribution of xM - xB is a normal distribution. This can be calculated by subtracting the mean of the men (188mg/dl) from the mean of the boys (170mg/dl), which gives a mean of 18 mg/dl. To calculate the standard deviation, we first calculate the standard deviation of the men (41 mg/dl) and the boys (30mg/dl). Then we use the formula: standard deviation of xM - xB = sqrt(standard deviation of men^2 + standard deviation of boys^2). This gives us a standard deviation of 35.4mg/dl. To calculate the probability of getting a difference in sample means xM - xB that's less than 0 mg/dl, we use the z-score formula: z = (x-mean)/standard deviation. Plugging in x = 0, mean = 18 and standard deviation = 35.4 gives us a z-score of -0.5197. Using a z-table, this gives us a probability of 0.3085. It should not be a surprise if the sample mean cholesterol for the 14 - year - olds exceeds the sample mean for the men as the mean cholesterol level for the boys is lower than the men.

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The angle measures are given as follows:

The sketch is given by the image presented at the end of the answer.

We have a right triangle, hence one of the angle measures is given as follows:

90º.

The sum of the measures of the internal angles of a triangle is of 180º, hence the sum of the remaining measures is given as follows:

90º, as 180 - 90 = 90.

The smallest angle is one fifth the medium size angle, hence:

x + 5x = 90

x = 15.

The measures are:

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The value of the variable x is 53.

The measures of the angles include the following:

m∠P = 96°.

m∠Q = 95°.

m∠R = 53°.

In Mathematics, a triangle can be defined as a two-dimensional (2D) geometric shape that comprises three (3) sides, three (3) vertices and three (3) angles only.

Furthermore, the sum of the angles in a triangle is equal to 180. This ultimately implies that, we would sum up all of the angles as follows;

m∠P + m∠Q + m∠R = 180

Substituting the given parameters into the mathematical expression, we have the following;

(2x - 10)° + (x + 42)° + x° = 180°

4x - 32° = 180°

4x = 180° + 32°

x = 53.

For the measure of the angles, we have:

m∠P = (2(53) - 10)° = 96°.

m∠Q = (53 + 42)° = 95°.

m∠R = 53°.

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Answer The total rental charge for Sandy's trip will be $601.05 (six hundrend one dollars and five cents)

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Step-by-step explanation:

Answer:

Step-by-step explanation:

A perfect cube monomial is a monomial with a cube root that is a monomial. In other words, it is a monomial, A, such that there exists a monomial, B, where B × B × B = A.

Answer:

250 adult tickets were sold

Step-by-step explanation:

Let A represent the number of adult tickets sold and  S the number of student tickets sold

Total tickets sold = A + S = 444

56 fewer student tickets than adult tickets means
S = A - 56

Substituting for s in the first equation,

A + S = 444

A + (A - 56) = 444

2A - 56 = 444

Add 56 to both sides:
2A - 56 + 56 = 444 + 56

2A = 500

A = 250

So 250 adult tickets were sold

Answer:

B

Step-by-step explanation:

Start with 200    in 5 mins  = 100     then cut in half for another 5 min = 50

   another 5 min cut in half again = 25

The initial value for Marcos to the grand canyon is 400 miles.

Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.

Both linear equations with one variable and those with two variables exist.

Given Marco is driving to the grand canyon. His distance from the Grand Canyon decreases by 150 mi every 3h.

We know that every 3 hours that distance decreases by 150 miles, then the distance decreased by the hour is:

150mi/3h = 50mi/h

since the distance is reducing we can take the rate of change negative,

so rate of change = -50 mi/h

and y = distance from the Grand Canyon

x = the number of hours he drives,

so the equation of slope is y = mx + c

where m = rate of change and c is the initial value at x = 0

m = -50 mi/h

y = -50x + c

And we know that after 4 hours the distance is 200 miles, then we can replace that to get:

200mi = (-50 mi/z)*4h + c

200mi = -200mi + c

200mi + 200mi = c  =400mi

equation is y = -50x + 400

Hence the initial value is 400 miles.

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