Callie spent 1 1/2 hours outside with her sister. They spent half of that time jumping on the trampoline. How long did they jump on the trampoline?

The given graph represents the function with the equation f(x) = 12(x - 1)/[(x + 2)(x - 3)].

To write an equation for the function, we first observe that the vertical asymptotes occur at x = -2 and x = 3, indicating that the denominator (Q(X)) has roots at x = -2 and x = 3. Hence, the equation of the denominator is:

Q(X) = (x + 2)(x - 3)

Furthermore, from the graph, we can see that the zero of the function occurs at x = 3, indicating that the numerator (P(X)) has the factor (x - 3). Let a be the leading coefficient of P(X), so we can write:

P(X) = a(x - 1)(x - 3)

To find the value of a, we use the fact that the y-intercept of the function is (0, -2). Substituting x = 0 and f(0) = -2 into the equation of the function, we get:

-2 = a(-1)/[(2)(-3)]

Simplifying this equation gives us:

a = -2/(2*3) = -1/3

Substituting this value of a into the equation for P(X), we get:

P(X) = (-1/3)(x - 1)(x - 3)

Substituting the equations for P(X) and Q(X) into the formula for f(x), we get:

f(x) = (-4)(x - 1)/[(x + 2)(x - 3)]

Simplifying this equation gives us:

f(x) = -4(x - 1)/[(x + 2)(x - 3)]

Therefore, the equation of the function graphed in the figure is f(x) = -4(x - 1)/[(x + 2)(x - 3)].

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[tex]x^2-5x+22=7x+2\implies x^2-12x+20=0 \\\\[-0.35em] ~\dotfill\\\\ \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{-12}x\stackrel{\stackrel{c}{\downarrow }}{+20}=0 ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases} \\\\\\ (-12)^2-4(1)(20)\implies 144-80\implies 64[/tex]

The licenses have space for 6 characters.

We need to note that there are 26 alphabets and 10 numbers to pick from.

So, for the first character, any of the 26 alphabets can take this spot.

For the second character, 25 alphabets are now available for that space. (Since repetition is not allowed)

For the third character, 24 alphabets are available for that.

For the fourth character, any of the 10 numbers can take up that spot.

For the fifth character, only 9 numbers can take this spot now. (No repetition rule too)

For the sixth character, 8 numbers can take that spot.

So, mathematically, the number of license tags possible will be

[tex]\bold{26 \times 25 \times 24 \times 10 \times 9 \times 8 = \underline{11,232,000} \ possible \ license \ tags}[/tex]

Answer:

11,232,000

Step-by-step explanation:

26 choices for the first letter.

25 choices left for the second letter (can't repeat letters)

24 choices left for the third letter

26 x 25 x 24 = 15,600 = possible first 3 characters.

10 choices for the first number

9 choices left for the second number (can't repeat numbers)

8 choices left for the third number

10 x 9 x 8 = 720 = possible last 3 characters.

chatgpt

Answer:

below

Step-by-step explanation:

To show that (AB)^-1 = B^-1A^-1, we need to show that:

(AB)(B^-1A^-1) = (B^-1A^-1)(AB) = I

where I is the identity matrix.

Using the associative property of matrix multiplication, we can simplify the left-hand side of the equation as:

(AB)(B^-1A^-1) = A(BB^-1)A^-1 = AIA^-1 = AA^-1 = I

Similarly, using the associative property of matrix multiplication, we can simplify the right-hand side of the equation as:

(B^-1A^-1)(AB) = B^-1(A^-1A)B = B^-1IAB = BB^-1 = I

Therefore, we have shown that (AB)^-1 = B^-1A^-1.

To show this identity, we need to demonstrate that:

(AB)^-1 = B^-1 A^-1

To do this, we start by multiplying both sides of the equation by (AB):

(AB)^-1 (AB) = B^-1 A^-1 (AB)

On the left side, we know that (AB)^-1 is the inverse of AB. So we can simplify to:

I = B^-1 A^-1 (AB)

Where I is the identity matrix. We can now simplify the right side by rearranging the terms:

B^-1 A^-1 (AB) = B^-1 (A^-1 A) B

Since A^-1 A = I, we can simplify further to:

B^-1 (A^-1 A) B = B^-1 IB = B^-1 B = I

Therefore, we have shown that:

(AB)^-1 = B^-1 A^-1

Q.E.D.

The coordinates of all four vertices of quadrilateral BPQF, we can use the shoelace formula to find its area. The shoelace formula states that the area of a polygon with vertices (x1, y1), (x2,

A triangle is a closed, two-dimensional geometric shape that has three sides and three angles. It is one of the basic shapes in geometry and can be formed by connecting three non-collinear points. Triangles can be classified based on their side lengths and angles.

To solve this problem, we can use the fact that the area of a triangle is equal to half the product of its base and height. Let's first find the length of the sides of the square.

Since the area of the square is 1320 square centimeters, we can find its side length by taking the square root of 1320:

√(1320) ≈ 36.32

So the side length of the square is approximately 36.32 cm.

Next, let's find the coordinates of points E and F. Since E is the midpoint of AB, its coordinates are:

E = ((0+36)/2, (36+0)/2) = (18, 18)

Similarly, since F is a quarter point of BC close to point B, its coordinates are:

F = (36, (3/4)36) = (36, 27)

Now let's find the equation of line CE. Since we know two points on the line (C and E), we can use the point-slope form of a linear equation to find its equation:

y - y1 = m(x - x1)

where m is the slope of the line and (x1, y1) is one of the points on the line. We can find the slope of the line by taking the difference in y-coordinates and dividing by the difference in x-coordinates:

m = (0 - 36)/(36 - 18) = -2

Using the point-slope form with point C, we get:

y - 0 = -2(x - 36)

y = -2x + 72

Similarly, the equation of line DB can be found using points D and B:

m = (0 - 36)/(0 - 18) = 2

y - 0 = 2(x - 0)

y = 2x

Finally, we can find the coordinates of point P by solving the system of equations:

y = -2x + 72

y = 2x

Setting the two expressions for y equal to each other, we get:

-2x + 72 = 2x

4x = 72

x = 18

Substituting x = 18 into either equation, we get:

y = 2x = 36

So the coordinates of point P are (18, 36). To find the coordinates of point Q, we can use the fact that it lies on line DF. Since we know two points on the line (D and F), we can again use the point-slope form of a linear equation:

m = (27 - 0)/(36 - 0) = 3/4

y - 0 = (3/4)(x - 0)

y = (3/4)x

To find the intersection point of lines DF and CE, we need to solve the system of equations:

y = (3/4)x

y = -2x + 72

Setting the two expressions for y equal to each other, we get:

(3/4)x = -2x + 72

(11/4)x = 72

x = 64/11

Substituting x = 64/11 into either equation, we get:

y = (3/4)(64/11) ≈ 14.73

So the coordinates of point Q are (64/11, 14.73).

Hence, Now that we have the coordinates of all four vertices of quadrilateral BPQF, we can use the shoelace formula to find its area. The shoelace formula states that the area of a polygon with vertices (x1, y1), (x2,

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Triangles to total shapes because triangles= 5 total, and total shapes = 10. 5/10 simplifies to 1/2

Answer:

!

Step-by-step explanation:

Between 2226 and 3241 people are predicted to see an improvement in their baldness.

The expected amount of people expected to see improvement is obtained applying the proportions in the context of the problem.

The sample proportion is given as follows:

175/500 = 0.35.

Considering the margin of error, the bounds are given as follows:

Hence the amounts out of 7810 people are obtained as follows:

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Answer:

We can use the formula:

average speed = total distance / total time

To find the total distance, we add the distances for each event:

total distance = 1.5 km (swimming) + 40 km (cycling) + 1 km (running) = 42.5 km

To find the total time, we add the times for each event:

time for swimming = distance / speed = 1.5 km / 2.5 km/h = 0.6 hours

time for cycling = 40 km / 12 km/h = 3.33 hours

time for running = 1 km / 9 km/h = 0.11 hours

total time = 0.6 hours + 3.33 hours + 0.11 hours = 4.04 hours

Now we can calculate the average speed:

average speed = total distance / total time = 42.5 km / 4.04 hours ≈ 10.52 km/h

Therefore, Farhan's average speed for the entire competition is approximately 10.52 km/h.

R2, since the model only has one independent variable.

Find R2:

R^{2} = [tex]\frac{SSR}{SST}[/tex]

R^{2} = [tex]\frac{18.26347892}{26.24578040}[/tex]

R^{2} = 0.6958634

[tex]R^{2} = 0.6959}}[/tex]

Part 2 of 3:

Find Ra2:

[tex]R^{2}_{a}= 1-\frac{(1-R^{2}) \\times (n-1)}{n-k-1}[/tex]

where

n = 9

since dftotal = n -1 = 8

thus n = 8+1=9

and k = number of independent variables = 1

Thus

[tex]R^{2}_{a}= 1-\frac{(1-0.6958634 ) \\times (9-1)}{9-1-1}[/tex]

[tex]R^{2}_{a}= 1-\frac{0.3041366 \\times 8}{7}[/tex]

[tex]R^{2}_{a}= 1-0.3475846[/tex]

[tex]R^{2}_{a}= 0.6524154[/tex]

[tex]R^{2}_{a}= 0.6524}}[/tex]

Part 3 of 3:

R2, since the model only has one independent variable.

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Answer: Angle aeb

Step-by-step explanation: To be supplementary, the angle(s) must add up to 180. Angle AEB and angle DEA add to 180. Another way to find this is to find the angle that the angle is adjacent to in a straight line.

11. Angle KEF = 40°, Angle KEJ = 50°

12. Angle HEF = 115°, Angle KED = 140°

13. Angle JEF = 90°, Angle KEF = 90°

14.   Angle DEF = 180°

The study of points, lines, and shapes in two and three dimensions, as well as their properties and relationships, is the focus of the mathematical discipline known as geometry.

11.  Acute angles: Those angles are less than 90 degree angles.

     Angle KEF = 40°

     Angle KEJ = 50°

     Angle KEH = 75°

     Angle JEH = 25°

     Angle HED = 65°

12. Obtuse angles: Those angles are more than 90 degree angles.

     Angle HEF = 115°

     Angle KED = 140°

13. Right angle: Those angles are 90 degree angle.

     Angle JEF = 90°

     Angle KEF = 90°

14. Straight angle: Those angles make 180 degree angle.

     Angle DEF = 180°

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Answer:

Step-by-step explanation:

If cos a = 0.5, then sin a could equal √(1 - cos^2 a) = √(1 - 0.5^2) = √0.75 ≈ 0.866.

BGC or AGF

Adjacent means next to or adjoining something else and both of the mentioned angles are next to angle AGB.

There is a probability of 0.117 for at least 2 of the tiles to be vowel tiles.

Given that,

Probability that the tiles getting is a vowel tile is 30%.

P(V) = 30% = 0.3

That is there will be only 3 tiles out of 10 tiles which are vowels.

Out of 8, there will be 8 × 0.3 = 2.4 tiles which are vowels.

There would be either 2 or 3 vowels.

Probability that at least 2 of them is vowel tiles is,

Probability = (0.3)² + (0.3)³

                 = 0.117

Here,

The simulation which can be used to fairly represent the situation is,

Use a computer to randomly generate 8 numbers from 1 to 10. Each time 1, 2 or 3 appears, it represents a vowel tile.

Hence, the required probability is 0.117.

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The standard form of the hyperbola is:

(x-3)²/4 - (y+2)²/(64/35) = 1

Given vertices (1, -2) and (5, -2), the center is (3, -2) and the semi-major axis, a = 2.

The hyperbola is horizontal.

Using point (-3, 6), we find the semi-minor axis, b² = 64/35.

The standard form of the hyperbola is:

(x-3)²/4 - (y+2)²/(64/35) = 1

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Answer:

True, False, True, False, False (answered top to bottom)

Step-by-step explanation:

Based on the Tangent Chord Theorem:

The Tangent Chord Theorem states that if a chord and a tangent intersect at the point of tangency, then the measure of each angle formed is equal to half the measure of its intercepted arc.

Considering the given angles;

m∠1 = ¹/₂ AB

m∠1 = ¹/₂ ACB

m∠DEG = ¹/₂ * 308°

m∠DEG = 154°

m∠DBC =  ¹/₂ * (360 - 192)

m∠DBC = 84°

m XY = 360 - (151 * 2)

m XY = 56°

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Answer:

A.the annual premium can be found by multiplying the monthly premium by 12:

Annual premium = Monthly premium x 12

Annual premium = $65 x 12 = $780

b. The semiannual premium can be found by dividing the annual premium by 2:

Semiannual premium = Annual premium / 2

Semiannual premium = $780 / 2 = $390

c. To calculate how much less Ronald's semiannual payments would be if he dropped the optional collision insurance, we need to subtract the cost of the optional insurance from the semiannual premium:

Semiannual payment with optional insurance = Semiannual premium - $120

Semiannual payment with optional insurance = $390 - $120 = $270

Semiannual payment without optional insurance = Semiannual premium - $120

Semiannual payment without optional insurance = $390 - $120 = $270

Therefore, Ronald's semiannual payments would be $270 if he dropped the optional collision insurance.

The transformations that are applied to the graph of the function [tex]f(x) = 10^x[/tex] to produce the graph of the function [tex]g(x) = 3(10)^x - 2[/tex] is: C. a vertical dilation by a factor of 3 and a vertical shift down 2 units.

In Mathematics and Geometry, a transformation can be defined as the movement of a point from its initial position to a new location. This ultimately implies that, when a function or object is transformed, all of its points would also be transformed.

In Mathematics and Geometry, a dilation refers to a type of transformation which typically changes the size of a geometric object, but not its shape.

Based on the parent function (exponential), we can logically deduce that the exponential function g(x) can be produced by vertically stretching the parent function by a scale factor of 3, followed by a vertical shift down 2 units;

[tex]f(x) = 10^x[/tex]

[tex]g(x) = 3(10)^x - 2[/tex]

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The linear equations from the options given are expressed as: y = 5 and y = -2/5x + 4.

An equation is linear can be expressed in the following forms:

y = mx + b

x = b

y = b

In the above, y is the dependent variable and x is the independent variable, m = slope, and b = y-intercept.

Note that in a linear equation the exponent of the variable is always 1. On the other hand, a nonlinear equation has an exponent that is not equal to 1 and can be expressed as:

y = x²

Therefore, the equations that are linear are:

y = 5 and y = -2/5x + 4.

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Answer: 86 memory sticks should be purchased

Step-by-step explanation: If he needs to be a stick for each teacher, and there are 86 teachers, then he needs to be 86 sticks.

The number of triangles formed by triangle ABC is 1

In order to calculate the possible number of triangles ABC that can be made with the specified parts, it is essential to employ conditions for triangle congruence, which includes the Triangle Inequality Theorem.

If sides a, b, and angle A can create a valid triangle, then they must follow these requirements:

a + b > c

a + c > b

b + c > a

By replacing given values with assigned variables, we get:

38 + 77 > c,

38 + c > 77,

77 + c > 38.

After solving these inequalities, we determine that c should be

However, if the Triangle Inequality Theorem is not observed, an unfeasible configuration results. Therefore, it would be impossible to generate a triangle using the following: AB = 38, AC = 77, and angle BAC = 74 degrees.

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The prominent similar feature between f(x) and g(x) is the present y-intercept.

Since the discriminant is negative, there are no possible solutions to the given equation, which implies that f(x) has no x-intercepts.

It should be noted that to identify the y-intercept, we set x = 0 and evaluate f(x):

f(0) = 2(0)² - 3(0) + 4

f(0) = 4,

The y-intercept of f(x) can be determined as (0, 4).

Moreover, the function g(x) = 2x-4 is a power function having a negative exponent, which makes it a reciprocal function. It's domain comprises all real numbers but 0, while its range includes all real numbers excluding 0.

In order to obtain the x-intercept of g(x), we set g(x) equal to zero and solve for x:

0 = 2x-4

2x = 4

x = 2

Consequently, the x-intercept of g(x) is (2, 0), together with the y-intercept (0, -4).

The prominent similar feature between f(x) and g(x) is the present y-intercept. On the other hand, f(x) does not contain any x-intercepts, whereas g(x) has a single x-intercept. Also, f(x) is categorized as a second-degree polynomial function, whereas g(x) is classified as a reciprocal function.

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Therefore, the closest option to the calculated percentage is option (d) 44%.

Percent, denoted by the symbol "%", is a way of representing a fraction or proportion out of 100. It is used to express a part of a whole in terms of hundredths. For example, 50% is the same as 50 out of 100, which simplifies to 1/2 or 0.5 as a decimal. Percentages are commonly used in many areas such as finance, statistics, and everyday life to express changes, rates, or comparisons.

Here,

To calculate the percentage of customers who prefer cookies, we need to add up the number of customers who prefer cookies from each survey and divide it by the total number of customers who took the survey.

Number of customers who prefer cookies: 25 + 9 + 28 + 11 = 73

Total number of customers who took the survey: 17 + 11 + 20 + 13 + 25 + 8 + 9 + 30 + 26 + 11 = 170

Percentage of customers who prefer cookies: (73/170) x 100% ≈ 44%

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If "t" number of years the study started and "y" denote the forest cover area, then the exponential-function representing the relation between "y" and "t" is y = 1500 × [tex](0.9225)^{t}[/tex].

An "Exponential-Function" is a mathematical expression in which a constant (called the base) is raised to a variable power.

In this case, we want to find an exponential function that relates the area of a forest (y) to the number of years since the start of an environmental study (t).

We know that the forest is decreasing by 7.75% each year, which means that the area of the forest after "t years" will be some percentage of its original area.

The formula for an exponential function is y = a × [tex]b^{t}[/tex], where "a" = initial value and "b" = growth or decay factor.

In this case, the initial area of the forest is = 1500 km², and

The forest is decreasing by 7.75% each year.

So, the "decay-factor" is = 1 - 0.0775 = 0.9225,

The exponential function that relates the area of the forest (y) to the number of years since the start of the study (t) is:

⇒ y = 1500 × [tex](0.9225)^{t}[/tex],

Therefore, the required "exponential-function" is y = 1500 × [tex](0.9225)^{t}[/tex].

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The probability that Mason does not arrive at work by 9 a.m. is given as follows:

p = 17/50.

A probability is calculated dividing the desired number of outcomes by the total number of outcomes.

The outcomes relating to the bus not arriving on time are divided as follows:

Hence the probability is given as follows:

p = 1/4 x 4/5 + 7/10 x 1/5

p = 1/5 + 7/50

p = 10/50 + 7/50

p = 17/50.

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