The height of a tree at time t is given by a twice-differentiable function H, where H(t) is measured in meters and t is measured in years. Selected values of H(t) are given in the table above.
(a) Use the data in the table to estimate H'(6). Using correct units. interpret the meaning of H'(6) in the context of the problem.
(b) Explain why there must be at least one time t, for 2 < t < 10, such that H'(t) = 2.
(c) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the average height of the tree over the time interval 2 ≤ t ≤ 10.
(d) The height of the tree, in meters, can also be modeled by the function G, given by
G(x) = 100x/(1+x), where x is the diameter of the base of the tree, in meters. When the tree is 50 meters tall, the diameter of the base of the tree is increasing at a rate of 0.03 meter per year. According to this model, what is the rate of change of the height of the tree with respect to time, in meters per year, at the time when the tree is 50 meters tall?
(This was on the no-calculator section of the recently-released AP Calculus AB 2018 exam so I appreciate it if you tried to limit calculator usage)

For the differential equation dy/dx = (x - 2)e^-2y having initial condition y(2) = ln(2), the solution is e^2y = x² - 4x + 8.

What is differential equation?

Any equation with one or more terms and one or more derivatives of the dependent variable with respect to the independent variable is referred to as a differential equation.

The given differential equation is dy/dx = (x - 2)e^-2y.

The initial condition is y(2) = ln(2).

Then the integration of the equation will be -

1/(e^-2y) dy = (x - 2) dx

e^2y dy = (x - 2) dx

(e^2y)/2 = (x²/2) - 2x + C

The value of C at x = 2, y(2) = ln (2).

Then it is obtained that -

[e^(2 ln 2)]/2 = [(2)²/2] - 2(2) +C

e^(ln 2²)/2 = 4/2 - 4 + C

Simplifying the equation further -

2²/2 = (4 - 8)/2 + C

2 = -4/2 + C

C = 2 + 4/2

C = 4

Then the equation will be -

(e^2y)/2 = (x²/2) - 2x + C

(e^2y)/2 = (x²/2) - 2x + 4

e^2y = x² - 4x + 8

Therefore, the solution is e^2y = x² - 4x + 8.

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The exact fractions. 2 x2−4 dx −2 = 10.6666

What is Fractions?

Any number of equal parts is represented by a fraction, which also represents a portion of a whole. A fraction, such as one-half, eight-fifths, or three-quarters, indicates how many components of a particular size there are when stated in ordinary English.

A fraction is used to denote a portion or component of a whole. It stands for the proportionate pieces of the whole. Numerator and denominator are the two components that make up a fraction. The numerator is the number at the top, and the denominator is the number at the bottom

[tex]\int\limits^2_2 {|x^{2} -4|} \, dx[/tex]

Eliminate absolutes

[tex]|x^{2} -4| : -2\leq x\leq 2[/tex]

without absolutes is

[tex]\int\limits^2_2 {x^{2} +4} \, dx \\\int\limits^2_2 {-x^{2} } \, dx +\int\limits^2_2 {4} \, dx \\[/tex]

=(-x³/3 )²₋₂ +4x )-₂²

=(-1/3 ) ( 8+8) +4(2+2)

= -16/3 +16

=32/3

= 10.6666

So the exact fraction is [tex]\int\limits^2_2 {|x^{2} -4|} \, dx[/tex]

=10.6666

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Answer:

4.7

Step-by-step explanation: anything over 5 you round up

b) An approximate value for e is 2.71828,

c) The natural exponential function is eˣ

The number e is a mathematical constant that is closely related to the exponential function. It is one of the most important and widely used numbers in mathematics and science.

The number e is defined as the limit of the sequence of numbers

=> (1 + 1/n)ⁿ

as n approaches infinity.

In other words, it is the unique number that satisfies the equation

=> eˣ = lim(n→∞) (1 + x/n)ⁿ.

This equation states that as n increases, the value of the exponential function of x approaches eˣ.

The approximate value of e is 2.71828, but it can be calculated to an infinite number of decimal places.

The natural exponential function is a mathematical function that is defined by

=> eˣ

It is used to model real-world situations that involve exponential growth or decay, such as the growth of a population or the decay of a radioactive substance.

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Conjecture about the relationship between tan(0) and tan(90 – 0) is tan(90-θ) is equal to cot θ.}. Tan(-) is equal to -tan, as is known. Therefore, eqn1 can be expressed as follows: -tan(90-), which results in -cot.

The equivalent of the given angle (90 degrees) in radians can also be used to indicate a tan angle of 90 degrees (1.57079 . . .) 90 degrees = 90° (180°/90°) rad = /2 or 1.5707... Given that the tangent function is periodic, we can write tan 90° as tan(90° + n 180°), where n Z.

tan(90-θ) is equal to cot θ.}

Tan(-θ) = -tanθ

-tan(90-θ) =  -cotθ.

-cotθ.

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Time taken to complete 10.3 half lives is 10.3 × 122 s = 1257 s which is equal to for the number of 15 8 O nuclei in a given sample to decrease to a factor of 8 × 10⁻⁴of the initial value.

Now, According to the question:

Radioactive Decay:

The rate of decay of nuclei of a radioactive element is directly proportional to the number of radioactive nuclei present at that moment in the sample.

The equation which determines the disintegration of atoms is,

N = N₀ e⁻(λt)

where, N is the number of atoms undergoing decay

N₀ is the initial number of atoms present

λ is the rate constant

t is the time

For the number of 15 8 O nuclei in a given sample to decrease to a factor of 8 × 10⁻⁴ of the initial value, the number of half-lives decay took place be n which is given by,

2⁻ⁿ = 8 × 10⁻⁴

Taking log on both sides we have,

- n ln 2 = ln(8 × 10⁻⁴)

n = 10.3

Time taken to complete 10.3 half lives is 10.3 × 122 s = 1257 s which is equal to for the number of 15 8 O nuclei in a given sample to decrease to a factor of 8 × 10⁻⁴ of the initial value.

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The original number was 4 is the correct answer.

Let's call the original number "x".

According to the problem, "a number is decreased by seven" which means x - 7.

And then, the new number is multiplied by 3 to get an answer of -9. That means (x - 7) * 3 = -9.

We can simplify the equation to 3x - 21 = -9 and then add 21 to both sides to isolate the variable: 3x = 12.

Finally, we can divide both sides by 3 to find x: x = 4.

So, the original number was 4.

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Based on the definition of the geometric mean, the value of x is: 64.

The geometric mean of two numbers is the square root of the product of the two numbers.

In other words, if x and y are two numbers, the geometric mean of x and y is represented as √(xy).

If 16 is the geometric mean between 4 and x, then we have the following relationship:

√(4x) = 16

Squaring both sides:

4x = 256

Dividing both sides by 4:

x = 64

So the value of x is 64.

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The graph shows the discontinuity of the function at x = -1 and x = 1. Therefore, the values of x at which the function is not continuous are -1 and 1.

The graph shows that at x = -1 and x = 1, the graph has a sharp jump in the value of the function. This means that the function is not continuous at those points, and thus the values of x at which the function is not continuous are -1 and 1. The graph represents a mathematical function, and the concept of continuity is crucial in mathematical analysis. A function is considered continuous at a specific point if the value of the function at that point exists and the limit of the function as the input approaches that point is equal to the value of the function at that point. In this particular graph, the function experiences a sharp increase or decrease in its value at the points x = -1 and x = 1. This sudden change in the function's value indicates that the function is not continuous at these points, meaning that its value does not approach a single limiting value as the input approaches these points. Thus, the points -1 and 1 are the values of x at which the function is not continuous.

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The time required to paint the new cone is 2.9 minutes.

Let the radius of the original cone be r.

∴ the area of the base is :

πr² = 49 π in²

so r² = 49.

Hence, the radius, r = 7 in.

∵ the lateral height of the cone is 8 inches, the slant height can be calculated using the Pythagorean Theorem: a² = b² + c²

i.e., s² = r² + h²

s² = 7²+ 8² = 49 + 64 = 113

s = √(113) in.

Let the area of the base of the new cone be, A

∴ A = 2 × 49 π in² = 98 π in².

Then, the radius of the new cone is √(98) in = 7 in × √(2).

The slant height of the new cone can be calculated in a similar manner:

s² = r² + h²

s² = (7 × √(2))² + 8² = 49 × 2 + 64 = 162

s = √(162) in.

The new cone will require more time to paint because its radius and height are bigger. Since the rate of painting is constant, the time needed to paint a fresh cone is inversely related to its surface area. Since the new cone's surface area is related to its slant height, the amount of time needed to paint it is equal to the new cone's slant height divided by the original cone's slant height. This means that it will take time to paint the new cone:

Time = 2.5 × √(162) ÷ √(113)

Time = 2.96 minutes

The time required to paint the new cone, rounded to the nearest tenth, is 2.9 minutes.

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Rounded to the nearest whole number, we would expect about 68 customers to pay with a credit card next week.

A whole number is a positive integer that does not have a fractional or decimal part. Whole numbers are the numbers we use for counting and for representing quantities that are not fractions or decimals. The set of whole numbers includes 0, 1, 2, 3, 4, 5, 6, and so on, with no limit on how high the numbers can go. Whole numbers do not include negative numbers or numbers with fractional or decimal parts.

To estimate the number of customers who will pay with a credit card next week, we can use the proportion of customers who paid with a credit card last week.

The total number of customers who paid with cash or a debit card is 77 + 27 = 104.

So the proportion of customers who paid with a credit card is 21/(104+21) = 0.1688 (rounded to four decimal places).

To estimate the number of customers who will pay with a credit card next week, we can multiply the expected total number of customers by this proportion:

0.1688 x 400 = 67.52

Rounded to the nearest whole number, we would expect about 68 customers to pay with a credit card next week.

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At B: 11:35 Bart will arrive back at the Railway Station for the final time

Let's call the number of circuits Bart completes "n".

Each circuit takes 5 minutes to travel between each stop and 5 stops in total, so each circuit takes 5 * 5 = 25 minutes.

The bus waits 10 minutes at the Railway Station between each circuit, so the total time for each circuit is 25 + 10 = 35 minutes.

Bart will complete 4 circuits, so the total time he spends on the bus tomorrow is 4 * 35 minutes = 140 minutes.

The first departure of the day from the Railway Station is at 09:15, so after 140 minutes of travel, Bart will arrive back at the Railway Station at 09:15 + 140 minutes = 11:35.

Therefore, the answer is B) 11:35.

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5 yards are equal to 15 feet. The solution has been obtained by using the unit conversion.

What is unit conversion?

The same feature is expressed in a different unit of measurement by using a unit conversion. For example, you may represent time in minutes rather than hours or distance in feet rather than miles. It is usual for measures to be given in one set of units, like feet, but to be requested in another set, like chains.

We are given 5 yards which are to be converted into feet.

We know that 1 yard = 3 feet

So,

⇒5 yards = 5 * 3

⇒5 yards = 15 feet

Hence, 5 yards are equal to 15 feet.

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limx→6[f(x)g(x)] = limx→6f(x) * limx→6g(x). Since limx→6f(x) = 15 and limx→6g(x) = -1, then limx→6[f(x)g(x)] = 15 * (-1) = -15. This means that the limit of the product of the two functions is -15.

We can use the limit laws of product to solve this limit.

limx→6[f(x)g(x)] = limx→6f(x) * limx→6g(x)

= 15 * (-1)

= -15

To solve this limit, we can use the limit laws of product. By the limit laws of product, the limit of a product is equal to the product of two limits. Therefore, limx→6[f(x)g(x)] = limx→6f(x) * limx→6g(x). Since limx→6f(x) = 15 and limx→6g(x) = -1, then limx→6[f(x)g(x)] = 15 * (-1) = -15. This means that the limit of the product of the two functions is -15.

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The most convenient type of loan Jane should consider obtaining based on her situation is an FHA loan.

FHA stands for Federal Housing Administration loan. This type of loan different from other loans allows those wanting to buy a house to request the loan with a very low down payment (about 3%). This makes it more accessible to people with limited incomes or who have not saved a lot of money to buy a house.

This is the best option for Janet because due to the low requirements, she can request the loan.

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The equation of a line that has a slope of -1/3 and y-intercept at (0, 1) is y = -x/3 + 1.

In Mathematics, the slope-intercept form of a line can be calculated by using this linear equation:

y = mx + c

Where:

From the information provided above, we have the following slope and y-intercept on the line:

Therefore, the required equation of line is given by:

y = mx + c

y = (-1/3)x + 1

y = -x/3 + 1

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The first six terms of the given sequence are 0, 3, 8, 15, 24, 35

A sequence is an enumerated collection of objects in which repetitions are allowed and order matters.

Given is a formula for a sequence aₙ = n²-1, we need to find the first six terms of the sequence,

Put n = 1 for first term,

a₁ = 1²-1

= 0

Put n = 2, for second term,

a₂ = 2²-1

= 3

Put n = 3, for third term,

a₃ = 3²-1

= 8

Put n = 4, for fourth term,

a₄ = 4²-1

= 15

Put n = 5, for fifth term,

a₅ = 5²-1

= 24

Put n = 6, for sixth term,

a₆ = 6²-1

= 35

Hence, the first six terms of the given sequence are 0, 3, 8, 15, 24, 35

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The stem-and-leaf plot of the given data is attached. The correct options regarding percentage distribution, the distribution skewness, and gaps within the data are A, E, and F.

A stem and leaf plot refers to a technique used to to present and organize quantitative data (discrete or continuous) in a graphical format. It is displayed as table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit). A stem and leaf plot can be used to visualize the shape of a distribution. The stem and leaf plot of the given data regarding the percentage loss of wetlands in the lower states is attached.

Based on the plot, it can be seen that:

Note: The question is incomplete. The complete question probably is: Wetlands offer a diversity of benefits. They provide a habitat for wildlife, spawning grounds for U.S. commercial fish, and renewable timber resources. In the last 200 years, the United States has lost more than half its wetlands. Environmental Almanac gives the percentage of wetlands lost in each state in the last 200 years. For the lower 48 states, the percentage loss of wetlands per state is given. i) Make a stem-and-leaf display of these data. (Use the tens digit as the stem and the ones digit as the leaf. Enter NONE in any unused answer blanks.) ii). How are the percentages distributed? Is the distribution skewed? Are there any gaps? (Select all that apply.) A) The percentages are concentrated from 20% to 60%. B) ii. The percentages are concentrated from 60% to 90%. C) There is a gap showing that none of the lower 48 states has lost from 60% to 69% of its wetlands. D) These data are strongly skewed left. E) These data are fairly symmetric, perhaps slightly skewed right. F) There is a gap showing that none of the lower 48 states has lost from 10% to 19% of its wetlands. G) There are no gaps in the data.

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The value of the equation is -46, the correct option is A.

An equation is an expression that shows the relationship between two or more numbers and variables.

A mathematical equation is a statement with two equal sides and an equal sign in between. An equation is, for instance, 4 + 6 = 10. Both 4 + 6 and 10 can be seen on the left and right sides of the equal sign, respectively.

Given;

The equation; a+16=-30

Now,

a=-30-16

a=-46

Therefore, the answer of equation will be -46

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Answer:

£4.80

Step-by-step explanation:

We know

A shop advertises 15% off

100% - 15% = 85%

So, £4.08 is the price at 85%

Find 1% by taking

4.08 divided by 85 = £0.048

Now find the price before the reduction by taking

0.048 times 100 = £4.80

So, it was £4.80 before the price reduction.

The transitive closure by arithmetic operations on the adjacency matrix is  Compute A⁰ + A¹ + ... + A⁽ⁿ⁻¹⁾ where + is standard addition (integers)

The transitive closure of a graph is the set of all possible paths from one vertex to another in the graph.

The best method to determine the transitive closure by arithmetic operations on the adjacency matrix is C.

Compute

=> A⁰ + A¹ + ... + A⁽ⁿ⁻¹⁾

where + is standard addition (integers).

The standard addition of integers is used because we are counting the number of paths between each vertex.

In this method, we start by computing A⁰, which is simply the adjacency matrix itself.

A¹ gives us the number of paths between each vertex that are one step long. A² gives us the number of paths between each vertex that are two steps long, and so on. We continue this process until we have computed

=> A⁽ⁿ⁻¹⁾

where n is the number of vertices in the graph.

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Step-by-step explanation:

If Sandra works for 42 hours per week and her weekly wage is £389.76, her hourly wage can be calculated as follows:

£389.76 / 42 hours = £9.28 per hour

When she receives a pay increase of 35p per hour, her hourly wage becomes:

£9.28 + 0.35 = £9.63 per hour

Her new weekly wage can then be calculated as:

£9.63 per hour * 42 hours = £404.66

So Sandra's new weekly wage is £404.66 after receiving a pay increase of 35p per hour.

A program is created that questions the user however many cookies they want to make and then displays how many cups of every ingredient are required to make that many cookies.

Writing code to support certain activities in a computer, application, or software program and giving them instructions on how to do is known as computer programming.

The given data for ingredients used in making cookie recipe.

user_cookies_input = float(input('Enter number of cookies: '))

cups_of_sugar = 1.5

cups_of_butter = 1.0

cups_of_flour = 2.75

number_of_cookies = 48

sugar_output = (user_cookies_input * cups_of_sugar) / number_of_cookies

butter_output = (user_cookies_input * cups_of_butter) / number_of_cookies

flour_output = (user_cookies_input * cups_of_flour) / number_of_cookies

print("You need" + int(sugar_output, '.2f') + "cups of sugar," + int(butter_output, '.2f') +

   "cups of butter, and" + int(flour_output, '.2f') + "cups of flour.")

Thus, a program is created that questions the user number of  cookies they want to make and then displays how many cups of every ingredient are required to make that many cookies.

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Answer:

33 miles

Step-by-step explanation:

You will use the Pythagoras' theorem ([tex]a^2+b^2=c^2[/tex])

Laura to her parents house is 'a', Her parents house to her grandparents house is 'c'

[tex]a^2+b^2=c^2\\c^2-a^2=b^2\\65^2-56^2=b^2\\4225-3136=b^2\\1089=b^2\\b=\sqrt{1089} \\b=33[/tex]

Laura is 33 miles from her grandparents house

Answer:

C

Step-by-step explanation:

15mpg

$1.09/gallon

10.5 per day * 22 days = 231 miles

will need 231/15 gallons of gas = 15.4gallons

cost = 15.4*1.09 = $16.79

Answer:

41

Step-by-step explanation:

x - 23 = 18  Add 23 to both sides

x - 23 + 23 = 18 +23

x = 41

Note can be repeated, but all the note cannot be the same to form an ordered five-note musical logo.

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Answer: A: 120

Step-by-step explanation:

When adding all sides of a triangle it should always equal 180 degrees so by adding 30+30=60 after you do 180-60=120 or the proper way is 30+30+x=180

60+x=180

-60     -60

x=120

An equation to represent how long it would take them working together to paint 20 rooms is m/15 + m/10 = 20.

In order to determine the amount of time it would take them working together to paint 20 rooms, we would have to assign variables to the amount of time taken by Maria to paint a room and the amount of time taken by Miek to paint a room, and then translate the word problem into an algebraic equation as follows;

Next, we would write an algebraic equation that models the combined work rate;

Combined work rate = 1/x + 1/y

(1 room/15 minutes) × m + (1 room/10 minutes) × m = 20 rooms

Substituting the given parameters into the algebraic equation, we have;

m/15 + m/10 = 20

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