What percent is 50% of 50% of x? Please help.

A rectangle has a a perimeter of 72 ft. The length and width are scaled by a factor 3. 5.

The perimeter of the new rectangle, which is the sum of its sides, is given by: P' = 2(l' + w')P' = 2(3.5l + 3.5w)P' = 2(3.5(l + w))P' = 2(3.5 x 36)P' = 2(126)P' = 252ft.

Therefore, the perimeter of the resulting rectangle is 252 ft.

Let the width of the rectangle be "w" and its length be "l".

Since the perimeter of a rectangle is the sum of the length of its sides, we can write:2(l + w) = 72ft(l + w) = 36ft

We can now find the ratio of the new length and width to the old ones: l' / l = 3.5 and w' / w = 3.5 .

The perimeter of the new rectangle, which is the sum of its sides, is given by:P' = 2(l' + w')P'

= 2(3.5l + 3.5w)P'

= 2(3.5(l + w))P' = 2(3.5 x 36)P'

= 2(126)P' = 252ft

Therefore, the perimeter of the resulting rectangle is 252 ft.

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The object weighs 24 pounds on Earth. The weight of an object on a certain planet is 2/5 of the weight on Earth. We know that an object weighs 9 3/5 pounds on the planet. So, we can use this information to find the weight of the object on Earth.

The equation to solve for w to find the object's weight on Earth in pounds is given by; w = 9 3/5 / 2/5 = 9.6 / 0.4 = 24

The object weighs 24 pounds on Earth. How to solve the equation?

The weight of an object on a certain planet is 2/5 of the weight on Earth. We know that an object weighs 9 3/5 pounds on the planet. So, we can use this information to find the weight of the object on Earth. To do this, we use the equation:

w = (2/5) * x

where w is the weight of the object on the planet and x is the weight of the object on Earth. We can substitute the values given into this equation to get:

w = (2/5) * x9 3/5 = (2/5) * x

Multiplying both sides by 5/2, we get:

x = 9 3/5 * 5/2x = 48/5

On simplification, we get: x = 9 3/5 pounds

So, the object weighs 24 pounds on Earth. This is our final answer.

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The set of vectors {(3, 2, 1, −6), (0, 6, 0, 2)} in [tex]R^n[/tex] is orthogonal.

To determine if the given set of vectors {(3, 2, 1, -6), (0, 6, 0, 2)} in [tex]R^n[/tex] is orthogonal, we must check if their dot product is zero. Orthogonal vectors are perpendicular to each other, and their dot product is an indicator of their orthogonality.
Let's take the dot product of the two given vectors:

A = (3, 2, 1, -6)
B = (0, 6, 0, 2)
Dot product (A•B) = (3×0) + (2×6) + (1×0) + (-6×2) = 0 + 12 + 0 - 12 = 0
Since the dot product of the two vectors is zero, we can conclude that the set of vectors is orthogonal.
Therefore, the correct answer is (a) orthogonal.

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The function to represent the mass of the sample after t years is

f(t) = 296.3895(0.4783)^t.

Given data: X is a radioactive isotope such that its mass decreases by 90% every year.

If an experiment starts out with 620 grams of Element X

We need to find a function to represent the mass of the sample after t years, where the daily rate of change can be found from a constant in the function.
Now, the percentage rate of change per day can be found as follows:

After one year, the mass decreases by 90%

So, at the end of the first year, the remaining mass

= 620 × 0.1

= 62 grams

Therefore, the percentage decrease in mass in one day

= (620 - 62) / 365

= 1.5 grams per day (approx.)

Thus, the percentage rate of change per day is

1.5 / 620

≈ 0.0024,

i.e., 0.24% per day

.A function to represent the mass of the sample after t years, where the daily rate of change can be found from a constant in the function can be represented by

Exponential function:

A = Ao * (1 - r) ^ t

Here, A = mass after t years

f(t)Ao = initial mass

= 620

r = percentage rate of change per day / 100

t = time in years

So, the function to represent the mass of the sample after t years is

f(t) = 620(0.1)^t or f(t)

= 620(0.9)^t

(As the mass decreases by 90% each year)

Hence, the required function is

f(t) = 620(0.9) ^ t

Round all coefficients in the function to four decimal places.

620 (0.9) ^ t = 620 (0.4783) ^ t

Hence, the required function is:

f(t) = 296.3895 (approx) * (0.4783) ^ t

Therefore, the function to represent the mass of the sample after t years is

f(t) = 296.3895(0.4783)^t.

Rounding to four decimal places, we get

f(t) ≈ 296.3895(0.4783)^t,

which is the required function.

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AIn problems 7 and 8, we need to find the solution of the given initial-value problem where x(0) = 1 and x'(0) = -3x. To solve this differential equation, we can separate the variables and integrate both sides. This gives us x(t) = e^(-3t/2). Thus, the solution of the initial-value problem is x(t) = e^(-3t/2) with x(0) = 1. The behavior of the solution as t approaches infinity is that x(t) approaches zero. This is because the exponential term e^(-3t/2) decays to zero as t becomes large.

To solve the given initial-value problem, we can use separation of variables. We start by separating the variables and get dx/x = -3/2 dt. Integrating both sides, we get ln|x| = -3t/2 + C, where C is a constant of integration. Solving for x, we get x = Ce^(-3t/2). We can then use the initial condition x(0) = 1 to find C. Plugging in x = 1 and t = 0, we get C = 1. Thus, the solution of the initial-value problem is x(t) = e^(-3t/2) with x(0) = 1.

To describe the behavior of the solution as t approaches infinity, we can look at the exponential term e^(-3t/2). As t becomes larger and larger, e^(-3t/2) approaches zero. This means that x(t) approaches zero as t approaches infinity. We can also see this by looking at the graph of the solution, which decays to zero as t becomes larger.

In conclusion, the solution of the initial-value problem x(0) = 1 and x'(0) = -3x is x(t) = e^(-3t/2). The behavior of the solution as t approaches infinity is that x(t) approaches zero. This is because the exponential term e^(-3t/2) decays to zero as t becomes large.

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Thus,  to determine the minimum dbar diameter to prevent fatigue failure, you need to consider the load cycles, material properties, stress range, structural design, and safety factor.

To determine the minimum reinforcing bar (dbar) diameter to ensure that fatigue failure will not occur, you need to consider the following factors:

1. Load Cycles: Fatigue failure typically occurs when a material is subjected to repeated cycles of stress. Analyze the expected number of load cycles and their magnitudes during the structure's service life.

2. Material Properties: The fatigue strength of the reinforcing bars depends on their material properties, such as yield strength, tensile strength, and ductility. Choose a dbar material that can withstand the anticipated stress cycles without causing fatigue failure.

3. Stress Range: Calculate the stress range (the difference between the maximum and minimum stress) the dbar will experience during the load cycles. This will help you assess the fatigue resistance of the material.

4. Structural Design: Optimize the structural design to minimize stress concentration and ensure uniform distribution of loads. This can help reduce the risk of fatigue failure.

5. Safety Factor: Apply an appropriate safety factor to account for uncertainties in material properties, load cycles, and structural design. This factor will help you determine a conservative minimum dbar diameter that reduces the risk of fatigue failure.

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The answer to your question is that the two general parts of the statistical mechanical expression for kp are the partition function and the reaction quotient.

The partition function is a fundamental concept in statistical mechanics that describes the distribution of particles among the available energy states in a system. It is used to calculate the probability of a system being in a particular state, and is a function of the temperature and the system's energy levels.

On the other hand, the reaction quotient is a measure of the relative amounts of reactants and products present in a chemical reaction at a given moment in time. It is calculated by dividing the concentrations (or partial pressures) of the products by the concentrations (or partial pressures) of the reactants, each raised to the power of its stoichiometric coefficient.

The statistical mechanical expression for kp therefore combines these two concepts, using the partition function to describe the distribution of energy states among the reactants and products, and the reaction quotient to determine the relative amounts of these species present in the reaction. The resulting expression provides a quantitative relationship between the equilibrium constant kp and the thermodynamic properties of the system, such as the temperature and the enthalpy and entropy changes associated with the reaction.

In summary, the two general parts of the statistical mechanical expression for kp are the partition function and the reaction quotient, which are used to describe the distribution of energy states and the relative amounts of reactants and products, respectively.

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The inverse Laplace transform of f(s) is f(t) = 10t + 7t^2/2 + 7t^3/3 + 80.125 t^4.

The inverse Laplace transform of f(s) = (3s - 7s^2 - 4s)/s^5 can be found by partial fraction decomposition. First, we factor the denominator as s^5 = s^2 * s^3 and write:

f(s) = (3s - 7s^2 - 4s) / s^5

= (As + B) / s^2 + (Cs + D) / s^3 + E / s^4 + F / s^5

where A, B, C, D, E, and F are constants to be determined. We multiply both sides by s^5 and simplify the numerator to get:

3s - 7s^2 - 4s = (As + B) * s^3 + (Cs + D) * s^2 + E * s + F

Expanding the right-hand side and equating coefficients of like terms on both sides, we obtain the following system of equations:

-7 = B

3 = A + C

0 = D - 7B

0 = E - 4B

0 = F - BD

Solving for the constants, we find:

B = -7

A = 10

C = -7

D = 49

E = 28

F = 343

Therefore, we have:

f(s) = 10/s^2 - 7/s^3 + 28/s^4 - 7/s^5 + 343/s^5

Using the inverse Laplace transform formulas, we can find the inverse transform of each term. The inverse Laplace transform of 10/s^2 is 10t, the inverse Laplace transform of -7/s^3 is 7t^2/2, the inverse Laplace transform of 28/s^4 is 7t^3/3, and the inverse Laplace transform of -7/s^5 + 343/s^5 is (343/6 - 7/24) t^4. Therefore, the inverse Laplace transform of f(s) is:

f(t) = l^-1 {f(s)}

= 10t + 7t^2/2 + 7t^3/3 + (343/6 - 7/24) t^4

= 10t + 7t^2/2 + 7t^3/3 + 80.125 t^4

Hence, the inverse Laplace transform of f(s) is f(t) = 10t + 7t^2/2 + 7t^3/3 + 80.125 t^4.

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Answer:

384 ft²

Step-by-step explanation:

The volume of the cylinder = π r²h

r = 3 ft

h = 8 ft

Let's solve

3 · 4² · 8 = 384 ft²

So, the volume of this cylinder is 384 ft²

The inequality that correctly compares Hailey's account values is: $117.39 > -$121.06.

To correctly compare the account values, we can use the inequality symbol.

Since Hailey has $117.39 in her savings account and -$121.06 in her checking account, the correct inequality to compare the values is:

Savings account value > Checking account value

Therefore, the correct inequality is:

$117.39 > -$121.06

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It will take approximately 4.02 years for the tube to have 200 million bacteria.

The exponential growth formula can be used to determine how long it will take for the tube to contain 200 million bacteria:

N = N₀ (1 + r)ⁿ

Where:

N is the final population size (200 million bacteria)

N₀ is the initial population size (50 million bacteria)

r is the growth rate (27% or 0.27)

n is the time in years

Putting the values,

200,000,000 = 50,000,000 (1 + 0.27)ⁿ

4 = (1 + 0.27)ⁿ

Taking the logarithm of both sides, we have:

log(4) = log((1 + 0.27)ⁿ)

n = log(4) / log(1 + 0.27)

n ≈ 4.02

Therefore, it will take approximately 4.02 years for the tube to have 200 million bacteria.

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The distance AD from the house to the school is given as follows:

AD = 10.63 km.

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

The distance in this problem can be represented by the hypotenuse of a right triangle, in which the sides are of 8 km and 3 + 4 = 7 km.

Hence the distance is given as follows:

d² = 7² + 8²

[tex]d = \sqrt{7^2 + 8^2}[/tex]

AD = 10.63 km.

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The required probabilities are: P (all three adults hold the opinion that there will be another housing bubble in the next four to six years) = 0.064 and P (none of the three adults hold the opinion that there will be another housing bubble in the next four to six years) = 0.216.

A)The probability of the first adult to hold the opinion that there will be another housing bubble in the next four to six years = P (E)

= 0.4

Therefore, the probability of the first adult not holding the opinion that there will be another housing bubble in the next four to six years = P (E')

= 1 - 0.4

= 0.6

Using the multiplication rule of probability,P (all three adults hold the opinion that there will be another housing bubble in the next four to six years) = P (E) × P (E) × P (E)

= 0.4 × 0.4 × 0.4

= 0.064 (3 decimal places)

B)The probability of one adult not holding the opinion that there will be another housing bubble in the next four to six years = P (E')

= 0.6

Using the multiplication rule of probability,

P (none of the three adults hold the opinion that there will be another housing bubble in the next four to six years)

= P (E') × P (E') × P (E')

= 0.6 × 0.6 × 0.6

= 0.216 (3 decimal places)

Therefore, the required probabilities are:

P (all three adults hold the opinion that there will be another housing bubble in the next four to six years) = 0.064 (3 decimal places)P (none of the three adults hold the opinion that there will be another housing bubble in the next four to six years) = 0.216 (3 decimal places)

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Answer:

9/100

Step-by-step explanation:

put it into ur calculator

A. Modelling Attendance for the first 5 years The decrease in attendance each year for the annual town international food festival is 20%. Therefore, the number of people attending the festival in year 1 is x.

From year 1 to year 2, the attendance will decrease by 20%. Therefore, the attendance for year 2 can be modeled as 0.8x.From year 2 to year 3, the attendance will again decrease by 20%. Therefore, the attendance for year 3 can be modeled as 0.8 × 0.8x = (0.8)²xFrom year 3 to year 4, the attendance will again decrease by 20%. Therefore, the attendance for year 4 can be modeled as 0.8 × (0.8)²x = (0.8)³xFrom year 4 to year 5, the attendance will again decrease by 20%.

Therefore, the attendance for year 5 can be modeled as 0.8 × (0.8)³x = (0.8)⁴xTherefore, the attendance for the first 5 years can be modeled as :[tex]x, 0.8x, (0.8)²x, (0.8)³x, (0.8)⁴x.B[/tex]. Attendance in 10  If the attendance decreases by 20% each year, then in 10 years, the attendance will decrease by 20% ten times. Therefore, the attendance in 10 years can be modeled as 0.8¹⁰x = 0.107x (rounded to three decimal places) or approximately 10.7% of the attendance in year 1.

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(a) To find the equation of the tangent line to h(x) = f(g(x)) at x = 2, we need to determine the derivative of h(x) and evaluate it at x = 2.

(b) To find F'(3) for F(x) = f(x)g(x), we need to calculate the derivative of F(x) and evaluate it at x = 3.

(a) The chain rule can be used to find the derivative of h(x). We first find the derivative of f(g(x)) with respect to g(x), which is f'(g(x)). Then, we multiply it by the derivative of g(x) with respect to x, g'(x). So, h'(x) = f'(g(x)) * g'(x). To find the equation of the tangent line at x = 2, we evaluate h'(x) at x = 2 and substitute the value into the point-slope form of a line using the coordinates (2, h(2)).

(b) To find F'(x), we apply the product rule, which states that the derivative of F(x) = f(x)g(x) is F'(x) = f'(x)g(x) + f(x)g'(x). We substitute x = 3 into F'(x) to find F'(3) by evaluating the derivatives of f(x) and g(x) at x = 3, and then performing the necessary calculations.

Note: The specific functions f(x) and g(x) and their derivatives are not provided in the given information, so their values would need to be determined or given to obtain the exact solutions for the equations of the tangent line and F'(3)

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The components of the velocity vector of the boat are approximately -37.62 km/hr in the x-direction (west) and -13.68 km/hr in the y-direction (south).

To find the components of the velocity vector of a boat moving at 40 km/hr in a direction 20° south of west, assuming north is in the positive y-direction.

Step 1: Convert the given angle to a standard angle (measured counterclockwise from the positive x-axis).

Since the boat is moving 20° south of west, we can find the standard angle by adding 180° to 20°.
Standard angle = 180° + 20° = 200°

Step 2: Calculate the x and y components of the velocity vector using trigonometry.

x-component = velocity * cos(standard angle)
y-component = velocity * sin(standard angle)

Step 3: Plug in the values and calculate the components.

x-component = 40 * cos(200°) ≈ -37.62 km/hr
y-component = 40 * sin(200°) ≈ -13.68 km/hr

In conclusion, the components of the velocity vector of the boat are approximately -37.62 km/hr in the x-direction (west) and -13.68 km/hr in the y-direction (south).

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In hex addition, values greater than 16 are mentally converted to their corresponding hexadecimal digits and then added together.

In hex addition, when the sum of two hexadecimal digits is greater than 15 (equivalent to 16 in base 10), it results in a carry.

To perform the addition mentally, you convert the carry value to its corresponding hexadecimal digit (A for 10, B for 11, C for 12, D for 13, E for 14, and F for 15) and add it to the next column. This process continues until there are no more carries left.

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Christa sliced the pyramid perpendicular to its base through one edge. The Option A .

A cross section means the view that shows what the inside of something looks like after a cut has been made across it. To determine how Christa sliced the cross section, let's consider the properties of a rectangular pyramid.

The rectangular pyramid has a rectangular base and triangular faces that converge at a single point called the apex. Since Christa sliced the pyramid through one edge perpendicular to its base, the resulting cross section would have the same shape as the base which is a rectangle.

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The number of students in Spring Lake Elementary School is given by 480.

The total number of students in Spring Lake Elementary School is given by = 600.

The percentage of students in Spring Lake Elementary School were absent on Monday is given by = 20 %.

So, the percentage of students in Spring Lake Elementary School were present on Monday is given by = (100 - 20) % = 80 %.

Thus, the number of students in Spring Lake Elementary School is given by = 80% of 600

= 600*80%

= 600 * (80/100)

= 6 * 80

= 480

Hence the number of students in Spring Lake Elementary School is given by 480.

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The line integral of F=xyi+yzj+xzk from (0,0,0) to (1,1,1) over the path C given by r=ti+t^2j+t^4k for 0≤t≤1 is 1/5.

To solve for the line integral, we first need to parameterize the curve. From the given equation, we have r(t) = ti + t^2j + t^4k.

Next, we need to find the differential of r(t) with respect to t: dr/dt = i + 2tj + 4t^3k.

Now we can substitute r(t) and dr/dt into the line integral formula:

∫[0,1] F(r(t)) · (dr/dt) dt = ∫[0,1] (t^3)(t^2)i + (t^5)(t)j + (t^2)(t^4)k · (i + 2tj + 4t^3k) dt

Simplifying this expression, we get:

∫[0,1] (t^5 + 2t^6 + 4t^9) dt

Integrating from 0 to 1, we get:

[1/6 t^6 + 2/7 t^7 + 4/10 t^10]_0^1 = 1/6 + 2/7 + 2/5 = 107/210

Therefore, the line integral is 107/210.

However, we need to evaluate the line integral from (0,0,0) to (1,1,1), not just from t=0 to t=1.

To do this, we can substitute r(t) into F=xyi+yzj+xzk, giving us F(r(t)) = t^3 i + t^3 j + t^5 k.

Then, we can substitute t=0 and t=1 into the integral expression we just found, and subtract the results to get the line integral over the given path:

∫[0,1] F(r(t)) · (dr/dt) dt = (107/210)t |_0^1 = 107/210

Therefore, the line integral of F over the path C is 1/5.

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The cdf for the random variable x is given by:
[tex]f_{x} (x)[/tex] = {   0             if x < -3
                               0.4             if -3 <= x < 5
                               0.8             if 5 <= x < 7
                                   1             if x >= 7   }

The probability of x = 5 is 0.

To find the probability of a specific value for a random variable, we use the probability mass function (pmf). The pmf is the derivative of the cumulative distribution function (CDF).
In this case, the cdf for the random variable x is given by:
[tex]f_{x} (x)[/tex] = {   0             if x < -3
                               0.4             if -3 <= x < 5
                               0.8             if 5 <= x < 7
                                   1             if x >= 7   }
To find the pmf, we take the derivative of fx(x) for each range of values:
P(x < -3) = 0 (no probability of x being less than -3)
P(x = -3) = [tex]f_{x}[/tex](-3) - [tex]f_{x}[/tex](-3-) = 0.4 - 0 = 0.4
P(-3 < x < 5) = [tex]f_{x}[/tex](5-) - [tex]f_{x}[/tex](-3) = 0.8 - 0.4 = 0.4
P(x = 5) = [tex]f_{x}[/tex](5) - [tex]f_{x}[/tex](5-) = 0.8 - 0.8 = 0
P(5 < x < 7) = [tex]f_{x}[/tex](7-) - [tex]f_{x}[/tex](5) = 1 - 0.8 = 0.2
P(x >= 7) = [tex]f_{x}[/tex](∞) - [tex]f_{x}[/tex](7-) = 0 - 1 = 0
Therefore, the probability of x = 5 is 0.

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Answer:

I don't know all of them but:

Question 3 is x=17. Because angles on a straight line sum 180 degrees.

(8x-15)+(3x+8)=180

x= 17

Question 5 is 78 degrees. Because the angle at the center is double the angle at the circumference.

The percentage of each is 0.66% , 1.65% , 2.97% , 37.21% , 37.48%, 20.1% respectively

The percentage of the total for each layer is calculated by dividing the depth of the layer in kilometers by the total depth

Percentage = (layer depth in km / total depth) × 100%

Crust= (40 / 6046) × 100 = 0.66%

Lithosphere = (100 / 6046) × 100 = 1.65%

Asthenosphere = (180/6046) × 100 = 2.98%

Mantle = (2250/6046) × 100 = 37.21%

Outer core = (2266/6046) × 100 = 37.48%

Inner core = (1210/6046) × 100  = 20.01%

The Depth in centimeters for each layer multiply the depth of the jar, 16.5 cm, by the percent you calculated for the crust

Crust = 0.66 × 16.5 cm =0.11 cm

Lithosphere = 1.65 × 16.5 = 0.27 cm

Asthenosphere = 2.98 × 16.5 = 0.49 cm

Mantle = 37.21 × 16.5 = 6.14 cm

Outer Core = 37.48 × 16.5 = 6.18 cm

Inner Core = 20.01 × 16.5 = 3.30 cm

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The question is incomplete the complete question is :

i. Divide the depth of the layer by the total depth. For example, to calculate the percentage of the total depth that the crust represents, divide 40 by 6,046.

ii. Write your answer in the Percent column.

iii. Repeat for the rest of the layers.

Use the calculator to determine the depth in centimeters for each layer. This is the depth of sand

you will put in your jar.

i. Multiply the depth of the jar, 16.5 cm, by the percent you calculated for the crust.

ii. Write your answer in the Centimeters column.

iii. Repeat for the rest of the layers.

Hello!

AB // CD => Thalès !

AO/OD = BO/OC = AB/CD

if BO = 24: (if not tell me in comments)

24/5 = AB/7.5

AB = 24 × 7.5 ÷ 5 = 36

Answer:

Since Ab||Cd

OB/AB=OC/CD

2/AB=5/7.5

AB=7.5×2/5

AB=3cm

Step-by-step explanation:

The first three nonzero terms of the Maclaurin series for f are 0, 0, and x^5/10.

To find the series, we use the Maclaurin series formula, which is a way to represent functions as an infinite sum of terms derived from their derivatives evaluated at a particular point. In this case, we evaluate the function's zeroth, first, and fifth derivatives at x=0 and obtain the first three nonzero terms of the series, which are 0, 0, and x^5/10.

The Maclaurin series is a powerful tool in mathematics and physics, and it is widely used in many areas such as calculus, differential equations, and quantum mechanics. By expressing functions as a series of terms, we can study their behavior and properties in greater detail, and make accurate predictions about their values for different inputs.

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When we are considering a random integer selected from the range from 2 to 10,000,000,000, there are 9,999,999,999 possible integers to choose from. Now, we need to determine how many of these integers are prime.

One way to approach this problem is to use the Prime Number Theorem, which states that the number of primes less than or equal to x is approximately x/ln(x). Using this theorem, we can estimate the number of primes less than or equal to 10,000,000,000 as:

[tex]\frac{10,000,000,000}{ln(10,000,000,000)} ≈ 455,052,511[/tex]

Therefore, there are approximately 455,052,511 prime numbers in the range from 2 to 10,000,000,000.

To find the probability of selecting a prime number, we need to divide the number of primes by the total number of integers in the range:

455,052,511/9,999,999,999 ≈ 0.0455

So, the chances of selecting a prime number from the range from 2 to 10,000,000,000 is approximately 0.0455 or 4.55%.

It is important to note that this is only an approximation based on the Prime Number Theorem and the actual number of primes in the range may differ slightly from this estimate. However, it gives us a good idea of the likelihood of selecting a prime number from this range.

The equivalent polar equation for the given rectangular equation x^2 + y^2 = 81 is r = 9. option (d) r = 9.

To find the equivalent polar equation for the given rectangular equation x^2 + y^2 = 81, we can follow these steps:

Step 1: Start with the given rectangular equation: x^2 + y^2 = 81.

Step 2: Convert x and y to polar coordinates using the conversions: x = r cos(θ) and y = r sin(θ).

Step 3: Substitute the polar coordinates into the rectangular equation:

(r cos(θ))^2 + (r sin(θ))^2 = 81.

Step 4: Simplify the equation:

r^2 cos^2(θ) + r^2 sin^2(θ) = 81.

Step 5: Use the trigonometric identity cos^2(θ) + sin^2(θ) = 1:

r^2(1) = 81.

Step 6: Simplify the equation:

r^2 = 81.

Step 7: Take the square root of both sides to solve for r:

r = 9.

Therefore, the equivalent polar equation for the given rectangular equation x^2 + y^2 = 81 is r = 9.

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The corresponding constant k in the exponential decay function is given by k = -(ln2)/T.

The exponential decay function for a radioactive substance can be expressed as:

N(t) = N₀[tex]e^{(-kt),[/tex]

where N₀ is the initial number of radioactive atoms, N(t) is the number of radioactive atoms at time t, and k is the decay constant.

The half-life, T, of the substance is the time it takes for half of the radioactive atoms to decay. At time T, the number of radioactive atoms remaining is N₀/2.

Substituting N(t) = N₀/2 and t = T into the equation above, we get:

N₀/2 = N₀[tex]e^{(-kT)[/tex]

Dividing both sides by N₀ and taking the natural logarithm of both sides, we get:

ln(1/2) = -kT

Simplifying, we get:

ln(2) = kT

Solving for k, we get:

k = ln(2)/T

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The derivation of the formula k = ln2/t gives us the half life of the isotope.

The amount of time it takes for half of a sample's radioactive atoms to decay and change into a different element or isotope is known as the half-life. It is a distinctive quality of every radioactive substance and is unaffected by the initial concentration.

We know that;

[tex]N=Noe^-kt[/tex]

Now if we are told that;

N = amount of radioactive substance at time = t

No = Initial amount of radioactive substance

k = decay constant

t = time taken

Then at the half life it follows that N = No/2 and we have that;

[tex]No/2 =Noe^-kt\\1/2 = e^-kt[/tex]

ln(1/2) = -kt

-ln2 = -kt

k = ln2/t

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