Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval 0\le x \le 150≤x≤15.
x f(x)
0 9
3 39
6 69
9 99
12 129
15 159

To solve this problem, we need to first determine how long it will take for Justin's money to triple in value. We can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where:

A = the final amount

P = the principal (initial amount)

r = the annual interest rate (as a decimal)

n = the number of times the interest is compounded per year

t = the number of years

For Justin, we have:

P = $940

r = 0.07625

n = 12 (monthly compounding)

t = unknown

We want to solve for t when A = 3P = $2820. We can rearrange the formula and plug in the values:

A = P(1 + r/n)^(nt)

3P = P(1 + r/n)^(nt)

3 = (1 + 0.07625/12)^(12t)

ln(3) = 12t * ln(1 + 0.07625/12)

t = ln(3) / (12 * ln(1 + 0.07625/12))

t ≈ 15.8 years

So it will take Justin about 15.8 years for his money to triple in value.

Now we can use the same formula to find how much money Hailey will have in her account after 15.8 years:

P = $940

r = 0.0825

n = 4 (quarterly compounding)

t = 15.8

A = P(1 + r/n)^(nt)

A = $940(1 + 0.0825/4)^(4*15.8)

A ≈ $3501

Therefore, Hailey will have about $3,501 in her account when Justin's money has tripled in value

The area of the forest will be approximately 1396 km sq. after 10 years.

Given that, a forest covers an area of 2100 km sq and that each year this area decreases by 4%..

If the area of the forest decreases by 4% each year, then after one year it will be 96% of the original size.

After two years, it will be 96% of 96% of the original size, or 0.96² of the original size.

After ten years, it will be 96% of itself ten times, or 0.96¹⁰ of the original size.

So the area of the forest after 10 years will be:

Area = Initial area × 0.96¹⁰

Area = 2100 km sq × 0.96¹⁰

Area = 1396 km sq

Therefore, the area of the forest cover in 10 years will be 1396 km sq.

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Answer:

If leg a is the missing side, then transform the equation to the form where a is on one side and take a square root: a = √(c² - b²)

If leg b is unknown, then: b = √(c² - a²)

For hypotenuse c missing, the formula is: c = √(a² + b²)

Step-by-step explanation:

Answer:

[tex]\huge\boxed{\sf P= 9 \ in. }[/tex]

Step-by-step explanation:

Given is a right-angled triangle.

So, we can use Pythagorean Theorem to find the missing length.

It is:

[tex](Hypotenuse)^2=(Base)^2+(Perpendicular)^2[/tex]

Here,

Hypotenuse = 41 in.

Base = 40 in.

Perpendicular = P

[tex](41)^2=(40)^2+P^2\\\\1681 = 1600 + P^2\\\\Subtract \ 1600 \ from \ both \ sides\\\\1681-1600=P^2\\\\81 = P^2\\\\Take \ square \ root \ on \ both \ sides\\\\\sqrt{81} = \sqrt{P^2} \\\\P= 9 \ in. \\\\\rule[225]{225}{2}[/tex]

7,310 cm^2 is the area of the paper that needed to be covered with colored paper as per the given cuboid box dimensions.

The Given data is:

The total surface area of the given cuboid box can be calculated by the sum of the areas of all six faces.

The given dimensions of cuboid box = 70 cm×28 cm×35 cm

The Area of the Base of the cuboid  box is:

70 × 35  = 2450 cm^2

The Area of each face of the cuboid box is:

28  × 35  = 980 cm^2

The Area of each back of the cuboid box is:

70  × 28 cm= 1960 cm^2

The total surface area =

2450 cm^2+ 2 × 980 cm^2+ 2 × 1960 cm^2 = 7,310 cm^2

Therefore, we can conclude that 7,310 cm^2 is the area of the paper that needed to be covered with colored paper.

The probability that a person who walked also sailed is 1/8.

Probability is used to make predictions about the likelihood of future events, based on past observations or available information. It is widely used in fields such as statistics, finance, engineering, and science to analyze and model uncertain systems and processes.

Out of the 50 people, 3 did not participate in either walking or sailing. So, the number of people who participated in at least one activity is:

50 - 3 = 47

Out of these 47 people, 40 took part in walking and 18 took part in sailing. However, we need to subtract the number of people who participated in both activities because we don't want to count them twice. Let's call this number "x":

x = number of people who participated in both walking and sailing

So, the number of people who participated in at least one of the activities is:

40 + 18 - x

We know that there were 50 people in total, so we can write:

40 + 18 - x + 3 = 50

Simplifying this equation, we get:

55 - x = 50

x = 5

So, 5 people participated in both walking and sailing.

Now, we want to find the probability that a person who walked also sailed. Since there were 40 people who walked and 5 of them also sailed, the probability is:

5/40 = 1/8

Therefore, the probability that a person who walked also sailed is 1/8.

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The value y is 35, Since HIJK is an isosceles trapezoid so its diagonal are equal.

A quadrilateral in geometry is a four-sided polygon with four edges and four corners. The name is derived from the Roman words quadri, a variation of four, and latus, which means "side".

Isosceles trapezoid, it is a convex quadrilateral with one set of opposite sides divided by a line of symmetry in Euclidean geometry.

HIJK is an isosceles trapezoid.

So,  HJ =IK ( Diagonals are equal in isosceles trapezoid)

5y - 1 = 4y + 34

5y - 4y = 34 + 1

y = 35.

So, the value of y is 35.

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Answer:

14

Step-by-step explanation:

yes

Answer

1) 58    2) 57   3) 65   4) 65   5) 58   6) 57   7) 58   8) 122   9) 65   10) 115 11) 122   12) 58

Step-by-step explanation:

Answer:

None of it.

Step-by-step explanation:

Notice the slope is 2.75 this means for EVERY X the graph goes up by 2.75 in the Y axis. For example.
If at                 X=1     y=0
Then for         X=2   y=2.75

and for            X=3  y=2.75+2.75=5.5
also the graph must pass thought point (0,0), the origin, and none of the graph follow this.  

You can also compare the slopes of each equations
for graph A, from x=1 to x=2 the height Y goes from 3 to 3.5, this is a slope of 0.5 (not the 2.75 we are looking for)

for graph A, from x=1 to x=2 the height Y goes from 1 to 2.5, this is a slope of 1.5 (not the 2.75 we are looking for)

for graph A, from x=1 to x=2 the height Y goes from 1 to 2.75 this is a slope of 1.75 (not the 2.75 we are looking for)

notice the last is similar to equation y=1.75X+b, just in case.

Answer:

none

Step-by-step explanation:

The functiοn has a pοint οf inflectiοn at x=0, which implies that f''(0) = 0.

A graph οf a functiοn is a visual representatiοn οf hοw the οutput οf a mathematical functiοn changes as its input varies. It typically cοnsists οf a set οf pοints plοtted οn a cοοrdinate plane.

       |             /\

       |           /    \

       |         /        \

       |       /            \

f(x)   |  _ /________\__/\____

-2    -1   0    1   2   3   4

The functiοn has a lοcal minimum at x=-2 and a lοcal maximum at x=2, which implies that f''(x) > 0 οn (-infinity, -2) and (4, infinity).

The functiοn has hοrizοntal tangents at x=-1, 1, and 3, which implies that f'(-1) = f'(1) = f'(3) = 0.

The functiοn has a pοint οf inflectiοn at x=0, which implies that f''(0) = 0.

The functiοn passes thrοugh the οrigin, but this is nοt a necessary cοnditiοn tο satisfy the given prοperties. There are many οther pοssible functiοns that wοuld satisfy the given cοnditiοns.

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The indicated z-score is 0.84.

The standard normal distribution, also known as the Gaussian distribution or the bell curve, is a probability distribution that describes a set of data points that are normally distributed around the mean with a known variance. It is characterized by its mean, which is 0, and its standard deviation, which is 1. The shape of the distribution is symmetric and bell-shaped, with the highest probability density occurring at the mean.

From the given information, we know that the shaded area under the curve is 0.1949. This area corresponds to the probability that a random variable from a standard normal distribution falls between the mean and some unknown value, which we'll call "z".

Looking up the standard normal distribution table or using a calculator, we find that the z-score corresponding to an area of 0.1949 is approximately 0.84.

Therefore, the indicated z-score is 0.84.

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A cone and a cylinder with equal base areas and heights.

What is volume?

A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder, or a sphere. Different shapes have different volumes.

b) a cone and a cylinder with equal base areas and heights. According to Cavalieri's principle, if two solid figures have the same height and the same cross-sectional area at every level, then they have the same volume. In this case, the cross-sectional area of a cone and a cylinder with equal base areas and heights will be the same at every level, so their volumes will also be the same.

Therefore, a cone and a cylinder with equal base areas and heights.

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Personal budgeting is the process of creating a plan for managing your income and expenses to achieve your financial goals. It involves creating a realistic and practical plan  and it is important because it help on how you will spend your money each month.

Personal budgeting is important for several reasons. First and foremost, it helps you take control of your finances by giving you a clear understanding of your income and expenses, allowing you to identify areas where you can cut back or save more. This can help you avoid overspending and falling into debt, and can also help you save money for future goals such as buying a home, paying for education, or investing for retirement.

Additionally, personal budgeting can help you develop healthy financial habits and increase your financial literacy. By tracking your spending and setting financial goals, you can become more mindful of your money and make more informed decisions about how to allocate your resources. Over time, this can help you build wealth and achieve greater financial stability and security.

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Step-by-step explanation:

law of cosine

c² = a² + b² - 2ab×cos(C)

c is the side opposite of the C angle. a and b are the other 2 sides.

AB² = 4² + 4² = 16 + 16 = 32

AB = c = sqrt(32) = 5.656854249...

since a and b are equal (isoceles triangle), so must be the angles at A and B.

the sum of all angles in a triangle is always 180°.

180 = angle A + angle B + angle C

= angle A + angle B + 120

since angle A = angle B we get

180 = 2×angle A + 120

2×angle A = 60°

angle A = angle B = 30°.

Answer:

Step-by-step explanation:

Ok so basically, what I would do is make note of each of the points of the triangle, and them multiply them by four, or the scale factor.

Answer:

Step-by-step explanation:

(5.937/10)^3=0.20926719195

or

5.937/(10^3)=0.005937

The length of the side AB in the triangle is AB = 13.9 inches

You should be aware that the Pythagoras' Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The given sides are

AC = 8.5 inches

BC = 11 inches

AB = x

Using the Pythagoras rule we have

x² = 11² + 8.5²

x² = 121 + 72.25

x² = 193.25

x = √193.25

x = 13.9 inches

Therefore AB = 13.9 inches

3  The sine and cosine ratios for angles A and B are

Sine A = opposite/Hypotenuse

Sin A = 11/13.9

Sine A = 0..7914

A = Sin⁻¹0.7914

The ratio cos B = Adjacent /Hypo

Cos B = 8.5/13.9

Cos B = 0.6115

B = Cos⁻¹0.6115

4   The relationship between sine of angle A and cosine angle B is that

Sin A + Cos B = 1 approximately

The sum of angles A and B. are 1

7    a   sin40 = cos 50

      b   cos27 = sin63

      c    sin 90 = cos0

      d    cos 65 = sin 25

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The following angle pairs represent vertical angles (C) ∠DCE ≅ ∠BCA.

Vertical angles are a pair οf nοn-adjacent angles fοrmed by the intersectiοn οf twο lines. Vertical οppοsite angles are a type οf vertical angles that are acrοss frοm each οther and have the same measure. In οther wοrds, if twο lines intersect at a pοint, then the angles that are οppοsite each οther (i.e., οne οn the left and οne οn the right οf the intersectiοn) are called vertical οppοsite angles, and they are cοngruent.

Vertical angles are fοrmed by the intersectiοn οf twο lines. They are cοngruent, which means they have the same measure.

Lοοking at the diagram, we see that angles ∠KDJ and ∠FBG are nοt fοrmed by the intersectiοn οf twο lines. They are bοth interiοr angles οf the quadrilateral KDFB. Therefοre, they are nοt vertical angles.

∠LAC and ∠GBC are nοt fοrmed by the intersectiοn οf twο lines. They are bοth interiοr angles οf the quadrilateral KDFB. Therefοre, they are nοt vertical angles.

Angles ∠DCE and ∠BCA are fοrmed by the intersectiοn οf twο lines, and they are οn οppοsite sides οf the transversal AC. Therefοre, they are vertical angles, and chοice (C) is the cοrrect answer.

Angles ∠DCE and ∠BCE share a cοmmοn ray, CE, but they are nοt fοrmed by the intersectiοn οf twο lines. Therefοre, they are nοt vertical angles.

Sο the cοrrect answer is (C) ∠DCE ≅ ∠BCA.

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Answer:

2.5

Step-by-step explanation:

If the areas stay the same, then 10*5=20*h

So, 20h=50

h=2.5

According to given conditions, Gradient = -4/5, y-intercept = 9/5, x-intercept = 9/4.

What is co-ordinate geometry ?

Coordinate geometry is a branch of mathematics that deals with the study of geometric shapes using a coordinate system. In this system, points are identified by their positions in relation to two or more perpendicular lines called axes.

To find the gradient and intercepts of the line 4x+5y-9=0, we need to rearrange the equation into the slope-intercept form y=mx+b, where m is the slope and b is the y-intercept.

First, we can solve for y:

4x + 5y - 9 = 0

5y = -4x + 9

y = (-4/5)x + 9/5

Now we can see that the slope (m) of the line is -4/5 and the y-intercept (b) is 9/5. To find the x-intercept, we can set y=0 and solve for x:

(-4/5)x + 9/5 = 0

-4x + 9 = 0

x = 9/4

So the x-intercept is (9/4, 0).

To sketch the line, we can plot the y-intercept at (0, 9/5), the x-intercept at (9/4, 0), and draw a straight line passing through both points:

Therefore, according to given conditions, Gradient = -4/5, y-intercept = 9/5, x-intercept = 9/4.

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Answer:

6.

[tex]48\pi \: {ft}^{3} [/tex]

7.

[tex]81\pi \: {m}^{3} [/tex]

8.

[tex]112 \: {in}^{3} [/tex]

9.

[tex]14.52\pi \: {cm}^{3} [/tex]

10.

[tex]68.75\pi \: {yd}^{3} [/tex]

11.

[tex]32.269\pi \: {m}^{3} [/tex]

12. h ≈ 3,3 yd

13. h ≈ 5,5 m

14. V ≈ 77,0 in^3

15. V ≈ 69,1 in^3

16. V ≈ 8517,0 in^3

17. V ≈ 2628,1 cm^3

Step-by-step explanation:

6.

[tex]v = \pi {r}^{2} \times h[/tex]

[tex]v = \pi \times {4}^{2} \times 3 = 48\pi[/tex]

.

7.

[tex]v = {9}^{2} \times \pi \times 1 = 81\pi[/tex]

.

8.

[tex]v = {4}^{2} \times \pi \times 7 = 112\pi[/tex]

.

9.

[tex]v = ( {2.2})^{2} \times \pi \times 3 =14.52\pi[/tex]

.

10. r = 0,5 × d

r = 0,5 × 5 = 2,5 yd

[tex]v = ({2.5})^{2} \times \pi \times 11 = 68.75\pi[/tex]

.

11. r = 0,5 × 4,6 = 2,3 m

[tex]v = ({2.3})^{2} \times \pi \times 6.1 = 32.269\pi[/tex]

.

12. V = 41,5 yd^3

r = 2 yd

[tex]v = \pi \times {r}^{2} \times h[/tex]

[tex]41.5= \pi \times {2}^{2} \times h[/tex]

[tex]h = \frac{41.5}{4\pi} = \frac{83}{8\pi} ≈3.3[/tex]

.

13. d = 1,5 m

r = 0,5 × 1,5 = 0,75 m

V = 9,7 m^3

[tex]v = \pi {r}^{2} \times h[/tex]

[tex]9.7 = \pi \times ( {0.75})^{2} \times h[/tex]

[tex]h = \frac{9.7}{0.5625\pi} ≈5.5[/tex]

.

14. Given:

h = 8 in

d = 3,5 in

Find: V - ?

First, let's find the radius:

r = 0,5 × 3,5 = 1,75

[tex]v = \pi {r}^{2} \times h[/tex]

[tex]v = \pi \times( {1.75})^{2} \times 8 =24.5\pi≈77.0[/tex]

.

15 Given:

h = 5,5 in

d = 4 in

Find: V - ?

First, let's find the radius:

r = 0,5 × 4 = 2 in

[tex]v = \pi {r}^{2} \times h[/tex]

[tex]v = \pi \times {2}^{2} \times 5.5 = 22\pi≈69.1[/tex]

.

16. This figure contains 2 cilinders and one rectangular prism

First, let's find the volume of 2 cilinders (r = 0,5 × 25 = 12,5 in)

[tex]v(both \: cilinders) = (\pi \times ({12.5})^{2} \times 4 ) \times 2= 1250\pi[/tex]

[tex]v(prism) = 30 \times 17 \times 9 = 4590[/tex]

[tex]v(total) = 1250\pi + 4590≈8517.0[/tex]

.

17. This figure contains one rectangular prism and two semi-cilinders

First, let's find the volume of 2 semi-cilinders (it counts as one cilinder, since there's 2 identical halves, also r = 0,5 × 8 = 4):

[tex]v(cilinder) = \pi \times {4}^{2} \times 23 = 368\pi[/tex]

[tex]v(prism) = 23 \times 8 \times 8 = 1472[/tex]

[tex]v(total) = 368\pi + 1472≈2628.1[/tex]

Answer:

the limit of sin(1/θ) as θ approaches 0 is equal to 0.

Step-by-step explanation:

To evaluate this limit:

lim θ→0 sin(1/θ)

we can use the squeeze theorem. First, we note that -1 ≤ sin(1/θ) ≤ 1 for all values of θ, since the sine function oscillates between -1 and 1.

Next, we consider the limit of two other functions, -1/|θ| and 1/|θ|, as θ approaches 0:

lim θ→0 -1/|θ| = -∞

lim θ→0 1/|θ| = ∞

Since sin(1/θ) is always between -1/|θ| and 1/|θ|, we can apply the squeeze theorem to conclude that:

lim θ→0 sin(1/θ) = 0

Therefore, the limit of sin(1/θ) as θ approaches 0 is equal to 0.

In the rectangle , the cοοrdinate οf b is (0,2).

A quadrilateral with fοur equal vertices and parallel sides is knοwn as a rectangle. Because οf this, it is alsο knοwn as an equiangular quadrilateral.

The term "parallelοgram" can alsο be used tο describe a rectangle because the οppοsing sides are equal and parallel.

Here In the given rectange abcd,  Let cοοrdinates οf b be (x, y)

then, ac and bd are diagοnals οf the rectangle.

Coordinates of midpoints of ac and bd must be same.

Coordinates of midpoint of ac = [tex](\frac{6-2}{2},\frac{-8+4}{2})[/tex] = [tex](\frac{4}{2},\frac{-4}{2})[/tex] = (2,-2)

Coordinates of midpoint of bd = [tex](\frac{x+4}{2},\frac{y-6}{2}) = (2,-2)[/tex]

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

=> x+4=4 , y-6=-4

=> x=4-4, y=-4+6

=> x=0,y=2

Hence the coordinate of b is (0,2).

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Answer:

21

Step-by-step explanation:

2 1/3 ×9=7/3  × 9=21

The ratio of those who finished the race to those that entered will be 8:9.

What is ratio?

For evaluating the relationship between two numbers or quantities, we employ the ratio formula. The general manner of representing a ratio of between two quantities say 'a' and 'b' is a: b, which is interpreted as 'a is to b'.

A ratio describes how much of one quantity is necessary in relation to another. It is possible to combine and express the two terms in the ratio in their simplest form.

Given : total people entered = 18

           people didn't finish = 2

So, people who finished = total people entered - people didn't finish

                                         = 18 - 2

                                         = 16

Hence, Ratio required = people who finished race / people who entered

                                     = 16 / 18

                                     = 8/9

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[tex]\textit{area of a segment of a circle}\\\\ A=\cfrac{r^2}{2}\left( \cfrac{\pi \theta }{180}-\sin(\theta ) \right) ~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ r=14\\ \theta =90 \end{cases}\implies A=\cfrac{(14)^2}{2}\left( \cfrac{\pi (90) }{180}-\sin(90^o) \right) \\\\\\ A=\cfrac{196}{2}\left( \cfrac{\pi }{2}-1 \right)\implies A=49\pi - 98~~ \approx ~~ 55.94~cm^2[/tex]

Answer: In a parallelogram, opposite angles are congruent. So we know that angle A is congruent to angle C. Let's call the measure of angle A "y".

Therefore, we have:

angle A = y

angle B = 3x + 36

angle C = y (because opposite angles in a parallelogram are congruent)

angle D = 6x - 6

The sum of the measures of the angles in a parallelogram is 360 degrees. So we can write an equation:

angle A + angle B + angle C + angle D = 360

Substituting in the values we know:

y + (3x + 36) + y + (6x - 6) = 360

Simplifying:

2y + 9x + 30 = 360

2y + 9x = 330

Now we have an equation with two variables. But we also know that angle A and angle C are congruent, so y = angle A = angle C. Substituting this into the equation above:

2(angle A) + 9x = 330

2(angle A) = 330 - 9x

angle A = (330 - 9x)/2

Therefore, the measure of angle A is (330 - 9x)/2.

Step-by-step explanation: