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XS Exponential growth and decay: word problems UKO
If a city with a population of 100,000 doubles in size every 28 years, what will the population
be 56 years from now?

Answer:

60 +24 = 84

Step-by-step explanation:jjjjjhjhjhj

6 x 4 +5 x 12

The interval on which the function f(x) = x^4 - 8x^2 + 9 is increasing can be expressed in interval notation as (-∞, -2) ∪ (2, ∞). The interval on which the function is decreasing can be expressed as (-2, 2).

To determine the intervals of increasing and decreasing, we need to examine the derivative of the function. Taking the derivative of f(x) with respect to x gives us f'(x) = 4x^3 - 16x. To find the intervals of increasing and decreasing, we need to analyze the sign of the derivative. The derivative is positive when x < -2 and x > 2, indicating an increasing function. The derivative is negative when -2 < x < 2, indicating a decreasing function.

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To find the length of the parametric curve described by the equations x = 9t^2 − 3t^3 and y = 3t^2 − 6t, we can set up an integral using the arc length formula. The curve starts at point (12, 9) and ends at point (0, 9), with several changes in direction along the way.

The length of a curve can be calculated using the arc length formula. For a parametric curve defined by x = f(t) and y = g(t), the arc length can be expressed as:

L = ∫[a,b] √[(dx/dt)^2 + (dy/dt)^2] dt.

In this case, we have x = 9t^2 − 3t^3 and y = 3t^2 − 6t. To find the length of the curve, we need to determine the interval [a, b] over which t varies.

From the given information, we can see that the curve starts at (12, 9) and ends at (0, 9). By solving the equation x = 9t^2 − 3t^3 for t, we find that t = 0 at x = 0, and t = 2 at x = 12. Therefore, the interval of integration is [0, 2].

To set up the integral, we calculate the derivatives dx/dt and dy/dt, and then substitute them into the arc length formula. Simplifying the expression inside the square root and integrating over the interval [0, 2], we can evaluate the integral to find the length of the curve.

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Answer:

Step-by-step explanation:

Answer:

6.6%

Step-by-step explanation:

To find the cumulative distribution function (CDF) of Z = X + Y, where X is uniformly distributed over (0,1) and Y is exponentially distributed with parameter lambda = 2, we can use the properties of independent random variables.

For 0 < a < 1, we have:

P(Z ≤ a) = P(X + Y ≤ a)

Since X and Y are independent, we can write this as:

P(Z ≤ a) = ∫∫ P(X ≤ x, Y ≤ a - x) dxdy

Since X is uniformly distributed over (0,1) and Y is exponentially distributed with parameter lambda = 2, we have their respective probability density functions (PDFs):

fX(x) = 1, 0 ≤ x ≤ 1

fY(y) = 2e^(-2y), y ≥ 0

Now, we can calculate the CDF of Z:

P(Z ≤ a) = ∫∫ P(X ≤ x, Y ≤ a - x) dxdy

= ∫∫ fX(x) * fY(y) dxdy, since X and Y are independent

= ∫∫ 1 *[tex]2e^(-2y)[/tex] dxdy, for 0 ≤ x ≤ 1 and y ≥ 0

Integrating with respect to x from 0 to 1 and with respect to y from 0 to a - x, we get:

P(Z ≤ a) = ∫[0,1]∫[0,a-x] 1 * 2[tex]e^(-2y)[/tex]dydx

= ∫[0,1] [[tex]-e^(-2y)[/tex]] [0,a-x] dx

= ∫[0,1] (1 - [tex]e^(-2(a-x)[/tex])) dx

Evaluating the integral, we have:

P(Z ≤ a) = [x - [tex]xe^(-2(a-x))[/tex]] [0,1]

= (1 - e^(-2a))

Therefore, the cumulative distribution function (CDF) of Z is:

P(Z ≤ a) = [tex](1 - e^(-2a)),[/tex] for 0 < a < 1

For 0 < a < ∞, the cumulative distribution function of Z remains the same:

P(Z ≤ a) = (1 - e^(-2a)), for 0 < a < ∞

Now, let's move on to the cumulative distribution function of T = X/Y.

P(T ≤ a) = P(X/Y ≤ a)

Since X and Y are independent, we can write this as:

P(T ≤ a) = ∫∫ P(X/y ≤ a) fX(x) * fY(y) dxdy

Using the given information that X is uniformly distributed over (0,1) and Y is exponentially distributed with parameter lambda = 2, we can substitute their respective PDFs:

P(T ≤ a) = ∫∫ P(X/y ≤ a) * 1 * [tex]2e^(-2y)[/tex]dxdy

= ∫∫ P(X ≤ ay) * 1 * [tex]2e^(-2y)[/tex]dxdy

Now, we need to determine the range of integration for x and y. Since X is between 0 and 1, and Y is greater than or equal to 0, we have:

0 ≤ x ≤ 1

0 ≤ y

Using these limits, we can calculate the CDF of T:

P(T ≤ a) = ∫[0,1]∫[0,∞] P(X ≤ ay) * 1 * [tex]2e^(-2y)[/tex] dydx

To evaluate this integral, we need to consider the range of values for ay. Since a can be any positive real number, ay can range from 0 to ∞.

P(T ≤ a) = ∫[0,1]∫[0,∞] P(X ≤ ay) * 1 * 2[tex]e^(-2y)[/tex] dydx

= ∫[0,1]∫[0,∞] (ay) * 1 * 2[tex]e^(-2y)[/tex] dydx, for ay ≥ 0

Integrating with respect to y from 0 to ∞ and with respect to x from 0 to 1, we have:

P(T ≤ a) = ∫[0,1]∫[0,∞] (ay) * 1 * 2[tex]e^(-2y)[/tex]dydx

= ∫[0,1] (2a / (4 + a^2)) dx

Evaluating the integral, we get:

P(T ≤ a) = (2a / (4 + [tex]a^2)),[/tex] for a > 0

Therefore, the cumulative distribution function (CDF) of T is:

P(T ≤ a) = (2a / (4 + [tex]a^2)),[/tex] for a > 0

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Answer:

The triangles are congruent because they are both exactly the same.

Answer:

length=2.5feet

Step-by-step explanation:

Volume =Base area * height

6/60=(4*length) * 6/6

10=4length

length=10÷4

=2.5feet

The length of the pyramid is 7.5 feet.

'A rectangular pyramid is a type of pyramid with the base shaped like a rectangle but the sides are shaped like a triangle. A pyramid usually has triangular sides but with different bases.'

According to the given problem,

Volume = 60 cubic feet

Height = 6 feet

Width of rectangular base = 4 feet

Let the length be x,

Volume =[tex]\frac{L * W * H}{3}[/tex]

⇒ 60 = [tex]\frac{x*6*4}{3}[/tex]

⇒ 180 = 24x

⇒ x = [tex]\frac{180}{24}[/tex]

⇒ x =[tex]\frac{45}{6}[/tex] = 7.5

Hence, we can conclude that the value of the length of the rectangular base is 7.5 feet.

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Answer:

the answer is the first one

Step-by-step explanation:

Explanation: be im smart

Function transformation involves changing the form of a function

A function that could represent the transformed function is function (c) [tex]f(x) = -\sqrt{\frac 12 x}[/tex]

The equation of the function is given as:

[tex]f(x) = \sqrt x[/tex]

The rule of horizontal stretch is:

[tex](x,y) \to (ax,y)[/tex]

Where:

[tex]0 < a < 1[/tex]

Take for instance:

[tex]a = \frac 12[/tex]

So, we have:

[tex]f(x) = \sqrt{\frac 12 x}[/tex]

Next, the function is reflected in across the x-axis.

The rule of this transformation is:

[tex](x,y) \to (x,-y)[/tex]

So, we have:

[tex]f(x) = -\sqrt{\frac 12 x}[/tex]

Hence, a function that could represent the transformed function is function (c)

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Answer:

they both = eachother

10x+9y=-17 = 10x-2y=16

Step-by-step explanation:

Answer:

6336 feet = 0.3476 nautical leagues

The statement Joe's salary of $35,060$⁢35,060 is 1.401.40 standard deviations above the mean is false.

To determine if the statement is true or false, we need to calculate the number of standard deviations Joe's salary of $35,060 is above the mean.

Given:

Mean salary = $28,600

Standard deviation = $4,400

Joe's salary = $35,060

To calculate the number of standard deviations above the mean, we can use the formula:

Number of standard deviations = (X - μ) / σ

Where:

X is the value we want to compare (Joe's salary)

μ is the mean

σ is the standard deviation

Plugging in the values, we have:

Number of standard deviations = (35,060 - 28,600) / 4,400

= 6,460 / 4,400

≈ 1.4727

Therefore, Joe's salary of $35,060 is approximately 1.4727 standard deviations above the mean, not 1.40.

The statement is false.

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The number of square feet-weeks used when producing a K flat of cucumbers for grafting is not provided.

To determine the number of sqftwks (square feet-weeks) used when producing a K flat of cucumbers for grafting that spent 6 weeks on a bench in the greenhouse, we need additional information.

The term "sqftwks" represents the product of the area in square feet and the duration in weeks. However, the specific values for the area and the duration of cucumber growth are not provided in the question.

To calculate the number of sqftwks, we would need to know the area occupied by the K flat of cucumbers and the time spent in the greenhouse. Without these specific values, it is not possible to determine the number of sqftwks used for cucumber production.

Therefore, based on the given information, we cannot calculate the number of sqftwks used when producing a K flat of cucumbers for grafting that spent 6 weeks on a bench in the greenhouse.

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The closest answer for each area is B. 0.052, D. 0.015, and C. 0.936, respectively.

(a) The area to the left of 32 on a N(45, 8) distribution. The area to the left of 32 on a N(45, 8) distribution is given by: P(Z < (32 - 45)/8)P(Z < -1.625)= 0.052, approximately. So, the closest answer is B. 0.052.

(b) The area to the right of 12 on a N(9.4, 1.2) distribution. The area to the right of 12 on a N(9.4, 1.2) distribution is given by: P(Z > (12 - 9.4)/1.2)P(Z > 2.166)= 1 - P(Z < 2.166)= 1 - 0.985= 0.015. So, the closest answer is D. 0.015.

(c) The area between 43 and 100 on a N(75, 15) distribution. The area between 43 and 100 on a N(75, 15) distribution is given by: P((43 - 75)/15 < Z < (100 - 75)/15)P(-1.5333 < Z < 1.6666)= P(Z < 1.6666) - P(Z < -1.5333)= 0.9525 - 0.0624= 0.8901. So, the closest answer is C. 0.936.

In conclusion, the closest answer for each area is B. 0.052, D. 0.015, and C. 0.936, respectively.

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The sample variance for the given data set is 3.6.

To calculate the sample variance, we follow a series of steps. First, we need to find the mean (average) of the data set. Adding up all the numbers and dividing by the total count gives us the mean, which in this case is (4+3+4+7+4+8)/6 = 30/6 = 5.

Next, we calculate the deviations of each data point from the mean. We subtract the mean from each data point to get the deviations: (4-5), (3-5), (4-5), (7-5), (4-5), and (8-5), which simplify to -1, -2, -1, 2, -1, and 3, respectively.

Then, we square each deviation to eliminate negative values:[tex](-1)^2[/tex], [tex](-2)^2[/tex], [tex](-1)^2[/tex], [tex]2^2[/tex], [tex](-1)^2[/tex], and [tex]3^2[/tex], which simplify to 1, 4, 1, 4, 1, and 9, respectively.

The next step is to find the sum of the squared deviations. Adding up all the squared deviations gives us 20.

Finally, we divide the sum of squared deviations by the total count minus 1 (n-1) to calculate the sample variance: 20/(6-1) = 20/5 = 4.

Rounding the sample variance to one decimal place, we get 3.6 as the final result.

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Answer:

(5,7) , (2, -3) , (2,4)

Step-by-step explanation:

your first number, the x, is the horizontal number. The second, y, is the vertical number. So if you look at A, the x is 5, and you then have to go up to 7 to reach the intercept

(a) The TQBF oracle used by M satisfies the condition for self-solvability.

It can handle formulas of length strictly shorter than the input length, ensuring that M's oracle queries are always strictly shorter than its input.

To show that TQBF (True Quantified Boolean Formula) is self-solvable, we need to demonstrate that there exists a polynomial-time oracle Turing machine (TM) M that can solve TQBF using an oracle for TQBF.

Assuming that formulas in TQBF are encoded into bit strings in a standard way, we can construct the TM M as follows:

On input x (the encoded TQBF formula), M starts by generating all possible truth assignments for the variables in the formula.

For each truth assignment, M constructs the corresponding quantified Boolean formula and queries the TQBF oracle with this formula.

If the oracle returns "true" for any truth assignment, M accepts x. Otherwise, if the oracle returns "false" for all truth assignments, M rejects x.

Hence, TQBF (True Quantified Boolean Formula) is self-solvable.

(b) If a language L is self-solvable, it implies that L is in PSPACE (polynomial space complexity class).

To prove that if L is self-solvable, then L is in PSPACE, we can show that a polynomial-time oracle TM M with oracle access to L can be simulated by a polynomial-space Turing machine.

Let M' be a polynomial-space Turing machine that simulates M with oracle access to L. Since L is self-solvable, M' can query the oracle on inputs shorter than its own input.

We can construct a polynomial-space Turing machine M'' that simulates M' without the need for an oracle. M'' uses its own polynomial space to simulate the computation of M'. Whenever M' queries the oracle on an input, M'' runs its own simulation for that input length using its available space.

Since M'' only requires polynomial space and simulates the behavior of M', it follows that L is in PSPACE.

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Answer:

Step-by-step explanation:

Total discount = 6×$0.90 - $4.86 = $0.54

discount each = $0.54/6 ≈ $0.09

$0.09/$0.90 = 0.10 = 10%

Answer:

I think it is 30

Step-by-step explanation:

Answer:

x = 30

Step-by-step explanation:

The angles shown are vertical angles

If you didn't know vertical angles are congruent

That being said we create an equation to solve for x

(because vertical angles are congruent) 104 = 3x + 14

now that we have created an equation we want to solve for x

Step 1 subtract 14 from each side

104 - 14 = 90

14 - 14 cancels out

now we have 90 = 3x

step 2 divide each side by 3

90/3=30

3x/3=x

we're left with x = 30

Answer:

117+x=180°(sum of straight line)

Step-by-step explanation:

x=180-117

x=63

Answer:

I think it's the second I e at the top

If the test statistic exceeds the critical value, we reject the null hypothesis and conclude that there is evidence to support the claim that the variance is greater than 15. Otherwise, if the calculated chi-square value is less than or equal to the critical chi-square value, we fail to reject the null hypothesis.

To test the hypothesis that the variance for the number of ounces of ice cream consumed per month is greater than 15, we can use a chi-square test for variance.

The null and alternative hypotheses for this test are as follows:

Null Hypothesis (H0): The variance for the number of ounces of ice cream consumed per month is equal to or less than 15.

Alternative Hypothesis (H1): The variance for the number of ounces of ice cream consumed per month is greater than 15.

Given a random sample of 23 people with a standard deviation of 4.4 ounces, we can calculate the sample variance as [tex]s^2 = (4.4)^2[/tex] = 19.36.

To perform the chi-square test, we calculate the test statistic as follows:

chi-square = (n - 1) × [tex]s^2[/tex] / [tex]\sigma^2[/tex]

where n is the sample size, [tex]s^2[/tex] is the sample variance, and [tex]\sigma^2[/tex] is the hypothesized population variance (15 in this case).

Substituting the values, we get:

chi-square = (23 - 1) × 19.36 / 15 = 29.44

Next, we compare this test statistic to the critical value from the chi-square distribution table with (n - 1) degrees of freedom (22 degrees of freedom in this case) at the given significance level (alpha = 0.05).

If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the variance for the number of ounces of ice cream consumed per month is greater than 15.

Otherwise, if the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis.

Performing the chi-square test with the given values, we compare the test statistic (29.44) to the critical value from the chi-square distribution table.

If the test statistic exceeds the critical value, we reject the null hypothesis and conclude that the variance for the number of ounces of ice cream consumed per month is greater than 15. Otherwise, we fail to reject the null hypothesis.

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Answer:

(7, 3 )

Step-by-step explanation:

Given the 2 equations

x - 3y = - 2 → (1)

x + 3y = 16 → (2)

Adding (1) and (2) term by term will eliminate the y- term

2x + 0 = 14

2x = 14 ( divide both sides by 2 )

x = 7

Substitute x = 7 into either of the 2 equations and solve for y

Substituting into (2)

7 + 3y = 16 ( subtract 7 from both sides )

3y = 9 ( divide both sides by 3 )

y = 3

solution is (7, 3 )

Answer:

2.9 litres

or

2 14/15 litres

Step-by-step explanation:

To determine the amount of water Sam drinks in an hour, divide the total amount drunk by the number of hours

Amount drank in an hour = total litres drunk / time

[tex]2\frac{1}{5}[/tex] ÷ [tex]\frac{3}{4}[/tex]

[tex]\frac{11}{5}[/tex] × [tex]\frac{4}{3}[/tex] = [tex]\frac{44}{15}[/tex] = 2[tex]\frac{14}{15}[/tex] litres

Answer:

8 ft

Step-by-step explanation:

Given:

Volume = 401.92

radius = 4 feet

Volume of cylinder: [tex]\pi[/tex]r^2h

3.14 * 4^2 * h = 401.92 cubic feet

3.14 * 16 * h = 401.92 cubic feet

50.24 * h = 401.92

h = 8 ft

Answer:

384 cubic inches (in^3)

Step-by-step explanation:

volume = length * width * height

So find the volume of the large rectangle of the prism, then find the volume of the small rectangle of the prism. Add the two volumes together and the sum is your final answer.

I would appreciate Brainliest, but no worries.

Answer:

180 degrees

Step-by-step explanation:

i think im not sure???????                                                                                                                      i hope im right

you put it in the question

Answer:

See Explanation

Step-by-step explanation:

Given

Let the bench be B and the flagpole be T.

So:

[tex]B = (0,10)[/tex] --- given

The flagpole is represented by the triangular shape labelled T.

So, we have:

[tex]T = (6,9)[/tex]

See attachment for the rectangular bench and the flagpole

From the attached image, the location of the other bench is:

[tex]B' = (0,-10)[/tex]

And the location of the other flagpole is:

[tex]T' = (-6,9)[/tex]

So, we have:

[tex]B = (0,10)[/tex] ==> [tex]B' = (0,-10)[/tex]

[tex]T = (6,9)[/tex] ==> [tex]T' = (-6,9)[/tex]

When a point is reflected from [tex](x,y)[/tex] to [tex](x,-y)[/tex], the transformation rule is reflection across x-axis.

So the rigid transformation that takes [tex]B = (0,10)[/tex] to [tex]B' = (0,-10)[/tex] is: reflection across x-axis.

When a point is reflected from [tex](x,y)[/tex] to [tex](-x,y)[/tex], the transformation rule is reflection across y-axis.

So the rigid transformation that takes [tex]T = (6,9)[/tex] to [tex]T' = (-6,9)[/tex] is: reflection across y-axis.

Answer:

0.625 : 1

Step-by-step explanation:

5 : 8

→ Divide both sides by 8 to get it to 1

(5 ÷ 8) : (8 ÷ 8)

→ Simplify

0.625 : 1