Ben made a sundial in his backyard by placing a stick with height, 6 inch straight into the ground, and marking the hours in the grass.

By answering the presented question, we may conclude that As a result, trigonometry the abbreviated phrase is: 8 sin(c+l)

Trigonometry is the field of mathematics that explores the connection between triangle side lengths and angles. The issue first originated in the Hellenistic era, during the third century BC, as a result of the use of geometry in astronomical investigations. The subject of mathematics known as exact techniques is concerned with certain trigonometric functions and their possible applications in calculations. Trigonometry contains six commonly used trigonometric functions. Their separate names and acronyms are sine, cosine, tangent, cotangent, secant, and cosecant (csc). Trigonometry is the study of triangle characteristics, particularly those of right triangles. As a result, geometry is the study of the properties of all geometric forms.

tan (7 sin(c) + 8 cos(c)) (l)

We may utilise the trigonometric identity to simplify this expression:

sin(l) / cos(l) = tan(l) (l)

When we insert this into the original phrase, we get:

sin(l) / cos(c) (7 sin(c) + 8 cos(c)) (l)

By increasing the numerator, we get:

8 cos(c) sin + 7 sin(c) sin(l) (l)

Now we can apply the trigonometric identities:

(1/2) sin(a) cos(b)

[sin(a+b) plus sin(a-b)]

(1/2) cos(a) sin(b)

[sin(a+b) minus sin(a-b)]

We can write using these identities:

7 sin(c) sin(l) + 8 cos(c) sin(l) equals 7 (1/2).

[sin(c+l) minus sin(c-l)] + 8 (1/2) [sin(c+l) plus sin(c-l)]

= (7/2)sin(c+l) + (8/2)sin(c-l) + (8/2)sin(c+l) + (7/2)sin(c-l) - (7/2)sin(c-l) = 8 sin(c+l)

As a result, the abbreviated phrase is:

8 sin(c+l)

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Answer:

2% of 800, which is 16

Answer: the equation have two solutions: 0.25 and 0.75

[tex]x=\frac{1}{4}[/tex]  or  [tex]x=\frac{3}{4}[/tex]

Step-by-step explanation:

Notice the terms [tex]x+x^2+x^3+x^4+...+x^n[/tex] is just a geometric series (since -1<x<1)

A geometric series is given by the form

[tex]1+x+x^2+x^3+x^4+...+x^n = \frac{1}{1-x}[/tex]
This is
[tex]x+x^2+x^3+x^4+...+x^n = \frac{1}{1-x} -1[/tex]
plugging in in the equation

[tex]-1+\frac{1}{x} +\frac{1}{1-x} -1=\frac{10}{3}[/tex]

grouping terms

[tex]\frac{1}{x} +\frac{1}{1-x} =\frac{16}{3}[/tex]
groupings the fractions

[tex]\frac{1-x+x}{(x)(1-x)}=\frac{1}{(x)(1-x)} =\frac{16}{3}[/tex]

this leads to an simplified quadratic equation since X cannot be zero (notice the initial 1/x term)

[tex]3=16(x^2-x)[/tex]
the solution for this quadratic equation is just

[tex]x=\frac{1}{4}[/tex]  or  [tex]x=\frac{3}{4}[/tex]

Answer:

  1/4 or 3/4

Step-by-step explanation:

You want the value of x in the infinite sum ...

  -1 +1/x +x +x² +x³ +...  = 10/3

for |x| < 1.

After the 2nd term, this looks like a geometric series with a common ratio of x. If we define the series sum as ...

  S = 1/x +1 +x +x² +x³ +...

we see that the equation of interest is ...

  S -2 = 10/3

The series sum is that of a geometric series with first term 1/x and common ratio x:

  S = (1/x)(1/(1 -x)) = 1/(x(1-x))

Substituting for S in the above, we have the equation ...

  1/(x(1-x)) -2 = 10/3

This resolves to a quadratic that will have 2 real roots:

  1/(x(1 -x)) = 16/3 . . . . add 2

  x(1 -x) = 3/16 . . . . . . . invert both sides

  x² -x +3/16 = 0 . . . . . .  subtract the left side expression

  (x -3/4)(x -1/4) = 0 . . . . . factor

Solutions are the values of x that make these factors zero:

  x = 1/4 or x = 3/4

Answer:

4/3 from x = 2 to x = 5.

Step-by-step explanation:

What is Lagrange mean value theorem?

Lagrange mean value theorem states that, if a function f is continuous over the closed interval [a, b] and differentiable over the open interval (a, b), then there must be at least one point c in the interval (a, b) where the slope of the tangent at the point c is equal to the slope of the tangent through the curve's endpoints, resulting in the expression f'(c) = {F(b) -F(a)}/(b-a)

According to the given graph,

At point x = 2,

F(2) = 3

At point x = 5,

F(5) = 7

Since the formula for the average rate of change of the function between x = a and x = b is,

The average rate of change = {F(b) -F(a)}/(b-a)

Here a = 2, b = 5 and F(2) = 3, F(5) = 7

Substitute the values in the formula,

So the average rate of change = (7 - 3)/(5 - 2) = 4/3.

Hence, the average rate of change of the function is 4/3.

Check the picture below.

[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x_1=2\\ x_2=5 \end{cases}\implies \cfrac{f(5)-f(2)}{5 - 2}\implies \cfrac{7-3}{5-2}\implies \cfrac{4}{3}[/tex]

Answer:

x = 10 (B)

Step-by-step explanation:

Use trigonometry:

[tex] \sin(30°) = \frac{5}{x} [/tex]

[tex] \frac{1}{2} = \frac{5}{x} [/tex]

Cross-multiply to find x:

[tex]x = 2 \times 5[/tex]

[tex]x = 10[/tex]

Step-by-step explanation:

3)

a. To find the height of the kite, we can use trigonometry. The sine function relates the opposite side (the height of the kite) to the hypotenuse (the length of string used) and the angle of elevation. Therefore, we can write:

sin(65°) = height/75

Solving for the height, we get:

height = 75 sin(65°) = 67.8 ft

Therefore, the kite is 67.8 feet high.

b. To find the distance between Joe and Jill, we can use trigonometry again. The tangent function relates the opposite side (the distance between Joe and Jill) to the adjacent side (the height of the bell tower) and the angle of depression. Therefore, we can write:

tan(angle of depression) = opposite/adjacent

tan(angle of depression) = Jill's height/210

Solving for the distance between Joe and Jill, we get:

distance = adjacent * tan(angle of depression)

distance = 210 * tan(angle of depression)

We need to know the angle of depression to solve for the distance, which is not given in the problem.

4)

a. In an isosceles triangle with two equal angles of 70°, the third angle must be:

180° - 70° - 70° = 40°

Since the triangle is isosceles, the height must be the perpendicular bisector of the base. Therefore, we can draw an altitude from the top vertex to the base, splitting the base into two equal segments. Let x be the length of each base segment. Then we can use trigonometry to find the height:

tan(70°) = height/x

height = x * tan(70°)

Since the height is given as 16 cm, we can solve for x:

16 = x * tan(70°)

x = 16/tan(70°)

Therefore, the length of each base segment is:

x = 16/tan(70°) = 6.12 cm

And the length of the base is twice the length of each segment:

base = 2x = 2(16/tan(70°)) = 12.25 cm

Therefore, the measures of the rest of the triangle are:

base = 12.25 cm

each equal angle = 70°

height = 16 cm

b. To determine if the ramp meets ADA standards, we need to find the angle of incline. The angle of incline is the angle between the ramp and the horizontal. We can use trigonometry to find this angle:

sin(angle of incline) = rise/run

sin(angle of incline) = 2/25

angle of incline = sin^(-1)(2/25)

Using a calculator, we get:

angle of incline ≈ 4.79°

Since the angle of incline is greater than the maximum allowable angle of 4.76°, the ramp does not meet ADA standards.

Answer:

  (f∘g)(1) = 2; (f∘g)'(1) = 2

  (f∘g)(2) = -2; (f∘g)'(2) = -2

Step-by-step explanation:

You want (f∘g)(x) and (f∘g)'(x) for x=1 and x=2 given the function values and derivatives in the table.

This composition means f(g(x)). The value is found by first determining the value of z = g(x), then using that to find the value of f(z).

For x=1, the value of g(1) is seen to be -2.

For x=-2, the value of f(-2) is seen to be 2.

This means f(g(1)) = 2.

For x=2, the value of g(2) is 0.

For x= 0, the value of f(0) is -2.

This means f(g(2)) = -2.

This is a little trickier, as you need to find the derivative of the composition:

  f(g(x))' = f'(g(x))·g'(x)

In the attached table, we have made a column for f'(g(x)) to help find this product.

For x=1, f'(g(1)) = f'(-2) = 1; and g'(1) = 2, so f'(g(1))g'(1) = 1·2 = 2 = (f∘g)'(1)

For x=2, f'(g(2)) = f'(0) = 2; and g'(2) = -1, so f'(g(2))g'(2) = 2(-1) = -2 = (f∘g)'(2)

Answer: The probability that two of the three CDs will be defective is approximately 0.01044 or 1.044%.

Step-by-step explanation: We can use the binomial distribution to solve this problem. Let X be the number of defective CDs in a sample of three. Then X follows a binomial distribution with parameters n = 3 and p = 6/100, where n is the sample size and p is the probability of a CD being defective.

The probability of getting exactly two defective CDs in a sample of three can be calculated using the binomial probability formula:

P(X = 2) = (3 choose 2) * (6/100)^2 * (94/100)^1

where (3 choose 2) = 3 is the number of ways to choose 2 defective CDs out of 3.

Simplifying this expression, we get:

P(X = 2) = 3 * (6/100)^2 * (94/100)

P(X = 2) = 0.01044

Therefore, the probability that two of the three CDs will be defective is approximately 0.01044 or 1.044%.

Answer: working on it

Step-by-step explanation:

Therefore , the solution of the given problem of expression comes out to be the correlation coefficient for this group is r = 0.9446.

Calculations like multiplication, variable splitting, joining, and presently removing are required. Combining them would result in the following: An equation, some statistics, and a mathematical formula. A declaration of truth is composed of values, components, mathematical processes like additions, subtractions, errors, and subdivisions as well as arithmetic formulas. Words and phrases can be evaluated and analysed.

Here,

You must use the following method to determine the r (Pearson correlation coefficient) for this sample:

=> r = (NΣXY - ΣXΣY) / √((NΣX² - (ΣX)²) (NΣY² - (ΣY)²))

where:

N stands for notes (in this case, 10)

Sum of each newborn's weight is equal to X.

Y = the measure of each infant

We can compute the following using the formula:

=> ΣX = 29.5

=> ΣY = 509

=> ΣXY = 1576.7

=> ΣX² = 97.95

=> ΣY² = 26757

When these numbers are added to the formula, we obtain:

=> r = (10 * 1576.7 - 29.5 * 509) / √((10 * 97.95 - (29.5)^2) * (10 * 26757 - (509)^2))

=> r = 0.9446

Since length and weight of newborns are strongly positively correlated, the correlation coefficient for this group is r = 0.9446.

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11 pieces can be cut frοm the 4 yards οf fabric.

An element οf a whοle is a fractiοn. The quantity is mathematically represented as a quοtient, where the numeratοr and denοminatοr are split in half. Bοth are integers in a simple fractiοn. A fractiοn can be fοund in either the numeratοr οr the denοminatοr οf a cοmplex fractiοn. The numeratοr οf an apprοpriate fractiοn is less than the denοminatοr.

1 yard = 36 inches (since 1 yard is equal tο 3 feet and 1 fοοt is equal tο 12 inches)

Sο, 4 yards οf fabric = 4 x 36 = 144 inches οf fabric.

If each piece is 13 inches lοng, we can find the number οf pieces by dividing the tοtal length οf fabric by the length οf each piece:

Number οf pieces = Tοtal length οf fabric / Length οf each piece

Number οf pieces = 144 / 13

Number οf pieces ≈ 11.08

Since we cannοt have a fractiοn οf a piece, we must rοund dοwn tο the nearest whοle number. Therefοre, 11 pieces can be cut frοm the 4 yards οf fabric.

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The table that represents a proportional relationship is the option (d) i.e. x: 6, 3, 0,  

y: -2, -1, 0

A general linear relationship is written as:

[tex]y = a*x + b[/tex]

Where a is called as the slope and b is called the y-intercept.

Proportional relationship is what we get when the y-intercept is 0, so we get an equation in the form:

[tex]y = a*x[/tex]

And in this case we can say, the slope is called the "constant of proportionality"

All proportional relationships of this property is given by , if we take x = 0 we get:

[tex]y = a*0 = 0[/tex]

So they always pass through the point (0, 0).

Now if we look at the given tables, the only option that passes through that point is the last table:

x: 6, 3, 0

y: -2, -1, 0

So that is the correct option.

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Solving for μ and σ simultaneously gives: μ = 3.2 hours (rounded to 1 decimal place) and σ = 0.426 hours (rounded to 3 decimal places)

What is Standard Deviation ?

Standard deviation is a statistical measure that describes the amount of variation or dispersion in a set of data. It measures how spread out the data is from its mean or average.

Let the mean of the population sleep time be μ and the standard deviation be σ.

From the given information, we know that the distribution of sample means of sleep time follows a normal distribution with mean μ and standard deviation σ/√40 (since each sample size is 40).

We are also given that 5% of the sample means are less than 3 hours and 5% of the sample means are more than 3.4 hours.

Using a standard normal distribution table, we can find the corresponding z-scores for these probabilities:

P(Z < z) = 0.05 when z = -1.645

P(Z > z) = 0.05 when z = 1.645

Now we can use the formula for z-score:

z = (X'  - μ) / (σ / √n)

where X' is the sample mean, n is the sample size (which is 40 in this case).

For the lower bound, we have:

-1.645 = (3 - μ) / (σ / √40)

For the upper bound, we have:

1.645 = (3.4 - μ) / (σ / √40)

Therefore, Solving for μ and σ simultaneously gives: μ = 3.2 hours (rounded to 1 decimal place) and σ = 0.426 hours (rounded to 3 decimal places)

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Answer:

22

Step-by-step explanation:

The time required for a radioactive isotope to decay to a certain percentage of its initial amount can be found using the following formula:

t = (t1/2 / ln(2)) * ln(N0/N1)

where:

t is the time elapsed since the release of the radioactive material

t1/2 is the half-life of the radioactive isotope (6 days in this case)

N0 is the initial amount of the radioactive isotope

N1 is the remaining amount of the radioactive isotope (14% of N0 in this case)

ln is the natural logarithm

We can solve for t by plugging in the given values:

t = (6 / ln(2)) * ln(1 / 0.14)

t ≈ 22.4 days

Therefore, the farmers needed to wait about 22.4 days to use the hay safely.

Answer:

[tex]{ \sf{ {b}^{3} = 8}} \: \: \\ { \sf{ {b}^{3} = {2}^{3} }} \\ { \sf{ {(b}^{3}) {}^{ \frac{1}{3} } = {( {2}^{ 3}) }^{ \frac{1}{3} } }} \\ { \sf{b ={ \boxed{ 2}}}}[/tex]

Answer:

b=2

Step-by-step explanation:

[tex]b^{3} = 8\\\\b^{3} = 2^{3} \\\\b=+2[/tex]

Answer:    2/15

Step-by-step explanation:
S: student wearing an sneakers
W: student wearing a white sneakers

Data given:

P(S) = 5/12
P(S∩W) = 8/25

so to find
P(W) = P(S)*P(S∩W) =  8/25* 5/12 = 2/15

This is 2/15 of the total students wears white sneakers

Answer:

2/15 of the student body normally wears white sneakers

Step-by-step explanation:

5/12

multiplied by

8/25

equals

40/300 = 4/30 =

after simplification

2/15

Answer: Right Picture

Step-by-step explanation:

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By using technology, the line of best fit include the following: C. y = 6.5x - 279.1.

In this scenario, the average daily temperature would be plotted on the x-axis of the scatter plot while the total ice cream sales would be plotted on the y-axis of the scatter plot.

On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.

From the scatter plot (see attachment) which models the relationship between the average daily temperature and total ice cream sales, a linear equation for the line of best fit is given by:

y = 6.5x - 279.1

In conclusion, the type of correlation between the variables is a strong positive correlation.

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Answer:

ŷ = 6.5x − 279.1

Step-by-step explanation:

Answer:

The cost of a specialty pizza is $19.49.

Step-by-step explanation:

To write a subtraction equation that could be used to find the cost c of a specialty pizza, we need to set up an equation that represents the relationship between the cost of a one-topping pizza and the cost of a specialty pizza.

The problem states that a one-topping pizza costs $6.50 less than a specialty pizza. Therefore, we can subtract $6.50 from the cost of a specialty pizza to get the cost of a one-topping pizza:

Cost of specialty pizza - $6.50 = Cost of one-topping pizza

We can rearrange this equation to solve for the cost of the specialty pizza:

Cost of specialty pizza = Cost of one-topping pizza + $6.50

Now, we can substitute the given value for the cost of a one-topping pizza:

Cost of specialty pizza = $12.99 + $6.50

Simplifying the expression, we get:

Cost of specialty pizza = $19.49

Therefore, the cost c of a specialty pizza is $19.49.

Hope this helps! I'm sorry if it doesn't. If you need more help, ask me! :]

Answer:

These distances show that AB, which is only 10 nautical miles apart, and AB are the closest pair of islands.

Step-by-step explanation:

We must first locate the three pairs of islands in order to establish which pair is nearest before determining the safest route for the ship.

Give the three islands the letters A, B, and C. The three island groups are designated as AB, AC, and BC. Finding the closest pair is necessary.

We can leverage the separation between the islands to do this. Assuming that the islands are separated by the following distances:

A and B are separated by 10 nautical miles.

A and C are separated by 15 nautical miles.

B and C are separated by 12 nautical miles.

These distances show that AB, which is only 10 nautical miles apart, and AB are the closest pair of islands.

Okay so you want to make all of the fractions have a common denominator. Think of the lowest number that 3 and 5 go into, the answer you should get is 15. So 15 is the common denominator.

Now we're going to make 1/3 have a denominator of 15. 1/3 of 15 would be 5. You get that by figuring out what you have to multiply 3 by to get 15 which is 5. Then you multiply 5 and 1 and you get 5.  The answer you will get will be 5/15 which is equal to 1/3.

Next we're going to make 4/5 have a denominator of 15. 4/5 of 15 would be 12/15. You can get this by figuring out what you have to multiply 5 by to get 15 which is 3. Then you take 3 and multiply it by 4 to get that 12 in 12/15. The answer 12/15 is equal to 4/5 of 15.

So now you have 5/15 + a = 12/15.

The denominator of the fraction that "a" will be is 15 because that is what we made the common denominator.

So what we have is 5/15 + x/15 = 12/15.

Think of what you have to add to 5 to get 12. (The answer is 7)

So a is equal to 7/15

Your final answer should be 5/15 + 7/15 = 12/15

Answer: 18/14, 27/21, 36/28

Step-by-step explanation:

By answering the presented question, we may conclude that as we know the sum of all angles is a triangle is 180; x = 46

A polygon is a triangle with four or more parts. It has a straightforward rectangular shape. Triangle ABC denotes a rectangle with the edges A, B, and C. Euclidean geometry produces a single plane and cube when the sides are not collinear. A triangle is a polygon if it contains three components and three angles. The corners are the points where a triangle's three edges meet. The angles of a triangle sum up to 180 degrees.

as we know the sum of all angles is a triangle is 180.

so, here,

20+114+x = 180

x = 180 - 134

x = 46

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The equivalent fraction of the given expression is: 2(y - 2x)(-5x + 3y)

The fraction expression is given as;

[(-2/x) + (5/y)]/[(3/y) - (2/x)]

Finding the common denominators and writing the numerators above common denominators gives:

[(2y - 4x)/xy]/[(-5x + 3y)/xy)]

Usually we divide a fraction by multiplying its' reciprocal and we have;

[(2y - 4x)/xy] * [(xy/(-5x + 3y)]

Thus, this gives us the expression:

(2y - 4x)/(-5x + 3y)

= 2(y - 2x)(-5x + 3y)

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Answer:

4+5/8

or

5/8+4

Step-by-step explanation:

"increased by" is adding, and "quotient" id dividing two numbers.

The mathematical expression '4 increased by the quotient of 5 and 8' is written as 4 + 5/8 in standard mathematical notation, meaning you first divide 5 by 8 and then add the result to 4.

The sequence of operations described can be converted into an expression in the following way: In terms of mathematics, the term 'increased by' refers to addition. The 'quotient of 5 and 8' refers to the result of dividing 5 by 8. Therefore, the expression can be written as '4 + 5/8'.

So, the mathematical expression '4 increased by the quotient of 5 and 8' becomes 4 + 5/8 in standard mathematical notation. This means you first divide 5 by 8 and then add the result to 4.

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a) The length of x is 5 cm. b)  the total surface area of the frustum is 61[tex]\pi[/tex]√802 + 16[tex]\pi[/tex] square cm (in terms of π).

A frustum has two bases, which are usually parallel to each other and are either circular or polygonal in shape. The height of the frustum is the perpendicular distance between the two bases, and the slant height is the distance between the apex of the frustum (the point where the original cone or pyramid was cut off) and any point on the perimeter of either base.

a. To find the value of x, we can use the similar triangles formed by the two cones. Let the radius of the small cone be y cm, then we have:

y/x = (35-7)/35 [Using the similarity of triangles]

Simplifying this expression, we get:

y = x(28/35) = 4x/5

Now, we know that the difference in the areas of the two circular bases of the frustum is equal to the area of the missing part. Using this fact, we can find the value of x as:

[tex]\pi[/tex](6²) - [tex]\pi[/tex](y²) = [tex]\pi[/tex](x²) - [tex]\pi[/tex]( (4x/5)² )

Simplifying this expression and solving for x, we get:

x = 5 cm

Therefore, the value of x is 5 cm.

b. The total surface area of the frustum can be calculated as the sum of the curved surface area of the small cone and the curved surface area of the frustum itself.

Curved surface area of the small cone = [tex]\pi[/tex](y²) = [tex]\pi[/tex](4²) = 16[tex]\pi[/tex]

Curved surface area of the frustum = [tex]\pi[/tex](6² + x²) × l

where l is the slant height of the frustum. To find the value of l, we can use the Pythagorean theorem:

l² = (35-7)² + (6-x)²

l² = 784 + (6-x)²

l = √[784 + (6-x)²]

Substituting the value of x, we get:

l = √[784 + (6-5)²] = √802

Therefore, the total surface area of the frustum is:

[tex]\pi[/tex](6² + x²) × √802 + 16[tex]\pi[/tex]

= [tex]\pi[/tex](6² + 5²) × √802 + 16[tex]\pi[/tex]

= 61[tex]\pi[/tex]√802 + 16[tex]\pi[/tex]

Hence, the total surface area of the frustum is 61[tex]\pi[/tex]√802 + 16[tex]\pi[/tex] square cm (in terms of [tex]\pi[/tex]).

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Answer:

P(x)=[tex](40000)( 0.98)^x[/tex]

A=40000    B=0.98
P(7)= 34700

Step-by-step explanation:

Notice the population can be modeled as:

P(x)=[tex]A B^x[/tex]

For X=0 P(0)=40000=A (Initial population)
So A=40000
For x=1 (one year after) P=39200

[tex]39200=40000 B^1[/tex]
solving for B
[tex]B=\frac{39200}{40000} =0.98[/tex]
So B=0.98

So the population can be modeled as

P(x)=[tex](40000)( 0.98)^x[/tex]

Now at 7 years:

P(7)=[tex](40000)( 0.98)^7[/tex]= 34725.02133
Needs to be rounded to the nearest hundred
This is 34700 bonobos (7 years after the study began)

The Volume of rectangular prism (V) = 20,825 [tex]cm^{3}[/tex]

A rectangular prism is a three-dimensional geometric shape that is formed by six rectangular faces joined together at right angles.

I think the question is, find out the volume of the rectangular prism in cubic centimeters? Given that the height is 85cm and area of base is 245cm².

We know that the formula for calculating the volume of a rectangular prism is as follows,

Volume of rectangular prism (V) = Area of base × height of the prism

Volume of rectangular prism (V) = 245cm² × 85cm

Volume of rectangular prism (V) = 20,825 [tex]cm^{3}[/tex]

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Answer:

Step-by-step explanation:

To calculate the payback period of the investment, we need to find out how long it will take for the company to recover the initial investment of 80000000 through the cash flows generated by the investment.

Step 1: Calculate the cumulative cash flow for each year.

Year 1: 25000000

Year 2: 25000000 + 25000000 = 50000000

Year 3: 50000000 + 20000000 = 70000000

Year 4: 70000000 + 20000000 = 90000000

Step 2: Determine the year in which the cumulative cash flow exceeds the initial investment.

Based on the calculations above, the cumulative cash flow exceeds the initial investment of 80000000 in Year 4.

Step 3: Calculate the payback period.

The payback period is the time it takes for the cumulative cash flow to equal the initial investment. In this case, the payback period is the end of Year 3 plus the portion of Year 4 needed to recover the remaining investment, which is calculated as follows:

80000000 - 70000000 = 10000000

10000000 ÷ 20000000 = 0.5

Therefore, the payback period for this investment is 3.5 years.

To confirm this result, we can also calculate the cumulative cash flow at the end of Year 3 and check that it is less than the initial investment, while the cumulative cash flow at the end of Year 4 exceeds the initial investment:

Year 1: 25000000

Year 2: 25000000 + 25000000 = 50000000

Year 3: 50000000 + 20000000 = 70000000 (cumulative cash flow at end of Year 3)

Year 4: 70000000 + 20000000 = 90000000 (cumulative cash flow at end of Year 4)

Since the cumulative cash flow at the end of Year 3 is less than the initial investment and the cumulative cash flow at the end of Year 4 exceeds the initial investment, we can confirm that the payback period is between Year 3 and Year 4, or 3.5 years.

Both f(x) and g(x) include domain values of [12,), and both functions increase over the interval (12, ∞).

A relationship between a set of inputs (the domain) and a set of potential outputs (the range) is known as a function. This relationship has the feature that each input is associated to exactly one output.

The notation f(x) is often used to express a function in mathematics, where f is the name of the function and x is the input value, also known as the argument of the function. When f(x) is calculated, it indicates the function's output value for the given input value x.

First, both functions have a domain that includes all values of x greater than or equal to 12, which is expressed as [12,) in interval notation.

Second, both functions increase over the interval (12, ∞), meaning that as x increases beyond 12, the values of the functions also increase without bound. This is because the exponential function 12^x and the square root function √(x - 12) both become increasingly steep as x increases, leading to a sharp increase in their values.

Therefore, Both f(x) and g(x) include domain values of [12,), and both functions increase over the interval (12, ∞).

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Answer:

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Step-by-step explanation:

is there a picture or something?