Step 6: x equals 20
Which justifies Step 6 of his work?

Position vector r(t) is given by

[tex]r(t) = (1/3)t^3 i + e^t j + e^{5t} k.[/tex]

Here's the step-by-step explanation:

1. Integrate each component of r'(t) with respect to t:
  ∫[tex](t^2) dt = (1/3)t^3 + C1[/tex] (for i-component)
  ∫[tex](e^t) dt = e^t + C2[/tex] (for j-component)
  ∫[tex](5te^{5t}) dt = e^{5t} + C3[/tex] (for k-component)

2. Apply the initial condition r(0) = i + j + k:
  r(0) = (1/3)(0)³ + C1 i + e⁰ + C2 j + e⁵ˣ⁰ + C3 k = i + j + k
  This implies that C1 = 1, C2 = 0, and C3 = 0.

3. Plug in the values of C1, C2, and C3 to find r(t):
  [tex]r(t) = (1/3)t^3 + 1 i + e^t j + e^{5t} k[/tex]

So, the position vector r(t) is given by [tex]r(t) = (1/3)t^3 i + e^t j + e^{5t} k.[/tex]

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The Partial F test is used to compare two nested linear regression models, where Model B is a more complex version of Model A. The null hypothesis of the test is that the additional variables in Model B do not have a significant impact on the dependent variable, while the alternative hypothesis is that they do.

To compute the Partial F test statistic, we need to first fit both models and obtain their respective residual sum of squares (RSS). Then, we can use the formula:

F = (RSS_A - RSS_B) / (p - q) * (RSS_B / (n - p))

where p is the number of variables in Model A (excluding the intercept), q is the number of additional variables in Model B (excluding those already in Model A), and n is the sample size.

The resulting F value follows an F-distribution with (q, n - p) degrees of freedom. We can then calculate the p-value by comparing this F value to the critical value of the F-distribution with the same degrees of freedom, using a significance level of 0.05.

If the p-value is less than 0.05, we reject the null hypothesis and conclude that Model B is a better fit than Model A. Otherwise, we fail to reject the null hypothesis and conclude that there is no significant difference between the two models.

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The 98% confidence interval for the true mean of exam 2 is approximately (62.81, 81.75).

To construct a 98% confidence interval for the true mean of exam 2, we will use the provided information:

Mean (μ) = 72.28
Standard deviation (σ) = 18.375
Sample size (n) = 25

First, we need to find the standard error of the mean (SE):
SE = σ / √n = 18.375 / √25 = 18.375 / 5 = 3.675

Next, we need to find the critical value (z) for a 98% confidence interval. The critical value for a 98% confidence interval is 2.576 (from the z-table).

Now we can calculate the margin of error (ME):
ME = z × SE = 2.576 × 3.675 ≈ 9.47

Finally, we can calculate the confidence interval:
Lower limit = μ - ME = 72.28 - 9.47 ≈ 62.81
Upper limit = μ + ME = 72.28 + 9.47 ≈ 81.75

So, the 98% confidence interval for the true mean of exam 2 is approximately (62.81, 81.75).

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The sequence above is an arithmetic sequence.

The common difference is -4.

In Mathematics and Geometry, the nth term of an arithmetic sequence can be calculated by using this expression:

aₙ =  a₁ + (n - 1)d

Where:

Next, we would determine the common difference as follows.

Common difference, d = a₂ - a₁

Common difference, d = 13 - 17 = 9 - 13

Common difference, d = -4.

Next, we would determine the common ratio as follows;

Common ratio, r = a₂/a₁

Common ratio, r = 13/17 ≠ 9/13

Common ratio, r = 0.7647 ≠ 0.6923

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The assertion "H: > 125" can be part of a legitimate statistical hypothesis when accompanied by a corresponding null hypothesis.

We will determine if the assertion "H: > 125" is a legitimate statistical hypothesis and explain why.

A statistical hypothesis is a statement about a population parameter that can be tested using sample data. There are two types of hypotheses: null hypothesis (H0) and alternative hypothesis (H1 or Ha). The null hypothesis is a statement of no effect, while the alternative hypothesis is a statement of an effect or difference.

In this case, the assertion "H: > 125" appears to be an alternative hypothesis, as it suggests that some parameter is greater than 125. However, for it to be a legitimate statistical hypothesis, it must be paired with an appropriate null hypothesis.

For example, if we were testing the mean weight of a certain species of animal, our hypotheses could be as follows:

- Null hypothesis (H0): The mean weight is equal to 125 (μ = 125)
- Alternative hypothesis (H1): The mean weight is greater than 125 (μ > 125)

With this pair of hypotheses, we can conduct a statistical test to determine whether the data supports the alternative hypothesis or not. In conclusion, the assertion "H: > 125" can be part of a legitimate statistical hypothesis when accompanied by a corresponding null hypothesis.

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The angle ACB is tan⁻¹(80/a), the range of tan⁻¹(x) is (0, 90) and the time taken to reach the shore is a/30

The measure of ACB can be calculated using the following tangent trigonometry ratio

tan(ACB) = Opposite/Adjacent

So, we have

tan(β) = 80/a

Take the arc tan of both sides

So, we have

β = tan⁻¹(80/a)

So, the angle is tan⁻¹(80/a)

Given that the angle is an acute angle

The range of tan⁻¹(x) for acute angles can be found by considering the values of the tangent function for angles between 0 and 90 degrees.

Since tan(0) = 0 and tan(90) is undefined, the tangent function takes on all positive values in this range.

So, the range of tan⁻¹(x) for acute angles is (0, 90) degrees.

Here, we have

Distance = a

Speed = 30 km/h

The time taken to reach the shore can be calculated using the formula:

time = distance / speed

Substituting the given values, we get:

time = a / 30 km/h

Simplifying this expression, we get:

time = a / 30 hours

Therefore, the time taken to reach the shore is a/30 hours, where a is the distance to the shore in kilometers.

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To express vector v as a linear combination of vectors v1, v2, and v3 and find Av, we need to know the components of vector v, and then we can set up and solve a system of linear equations to determine the coefficients c1, c2, and c3, and calculate Av using matrix multiplication.

In order to express vector v as a linear combination of vectors v1, v2, and v3, we need to know the components of vector v. The components of a vector represent its values along each coordinate axis or direction. Let's assume that the components of vector v are denoted as v_x, v_y, and v_z, representing its values along the x, y, and z axes respectively.

Given that, we can express vector v as a linear combination of vectors v1, v2, and v3 as follows:

v = c1 * v1 + c2 * v2 + c3 * v3

where c1, c2, and c3 are constants that represent the coefficients or weights of the respective vectors v1, v2, and v3 in the linear combination.

To find the coefficients c1, c2, and c3, we can set up a system of linear equations based on the components of vector v and the given vectors v1, v2, and v3. We can then solve this system of linear equations to determine the values of c1, c2, and c3.

Once we have the coefficients c1, c2, and c3, we can also calculate Av, which represents the vector resulting from the matrix multiplication of a matrix A (formed by stacking v1, v2, and v3 as columns) and the column vector containing c1, c2, and c3 as its elements.

In summary, to express vector v as a linear combination of vectors v1, v2, and v3 and find Av, we need to know the components of vector v, and then we can set up and solve a system of linear equations to determine the coefficients c1, c2, and c3, and calculate Av using matrix multiplication.

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Answer:

EG = 75°

Step-by-step explanation:

the secant- secant angle DFQ is half the difference of its intercepted arcs, that is

∠ DFQ = [tex]\frac{1}{2}[/tex] (DQ - EG) , substituting values

35° = [tex]\frac{1}{2}[/tex] (145 - EG ) ← multiply both sides by 2 to clear the fraction

70° = 145 - EG ( subtract 145 from both sides )

- 75 = - EG ( multiply both sides by - 1 )

EG = 75°

The temperature of the soup after 5 minutes is [tex]26 + 42.5(e^4 - 1)[/tex]degrees Celsius.

1. The given expression represents the accumulation function of the population of the beachside resort. It is the integral of the rate function r(t) over the time interval [2, t], where t is the current time in years. The value of the integral at t is added to the initial population of 4823 to get the current population. In other words, the expression represents the total number of residents that have moved into the resort from time 2 to time t.

So, the expression can be written as: [tex]4823 + \int 2t r(x) dx = 7635 + \int 2t r(x) dx[/tex]

2. To find how much the temperature increased between t = 0 and t = 10 minutes, we need to evaluate the integral of the rate function r(t) over the time interval [0, 10]. The value of the integral will give us the total increase in temperature during this time period.

So, the expression can be written as[tex]\int 0^{10} 34e^{0.8t} dt[/tex]

Simplifying the integral, we get[tex]: [42.5e^{0.8t}]0^{10} = 42.5(e^8 - 1)[/tex] degrees Celsius

Therefore, the temperature of the soup increased by[tex]42.5(e^8 - 1)[/tex]degrees Celsius between t = 0 and t = 10 minutes.

3. To find the temperature of the soup after 5 minutes, we need to evaluate the expression for the accumulation function of temperature at t = 5, given that the initial temperature is 26 degrees Celsius.

So, the expression can be written as:[tex]26 + \int 0^5 34e^{0.8t} dt[/tex]

Simplifying the integral, we get: [tex]26 + [42.5e^{0.8t}]0^5 = 26 + 42.5(e^4 - 1)[/tex] degrees Celsius

Therefore, the temperature of the soup after 5 minutes is[tex]26 + 42.5(e^4 - 1)[/tex]degrees Celsius.

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The correct answer to the above threads-based question Option is d. 1/6 inch.

The phrase "12 threads per inch" refers to the number of ridges or threads on the screw shaft for every inch of its length. This indicates that the space between two consecutive threads for a single thread is 1/12 inch.

When the screw turns two complete revolutions, it travels a distance equal to the screw pitch. The spacing between neighboring threads is specified as the screw pitch. Because there is just one thread in this scenario, the pitch is 1/12 inch.

When the screw completes two complete rotations, it advances into the nut a distance equal to the screw pitch, which is 1/12 inch. Since the answer choices are given in fractions, we can simplify 1/12 to 1/6 by dividing both the numerator and denominator by 2. Hence, the correct answer is d) 1/6 inch.

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The correct answer to the above threads-based question Option is d. 1/6 inch.

The phrase "12 threads per inch" refers to the number of ridges or threads on the screw shaft for every inch of its length. This indicates that the space between two consecutive threads for a single thread is 1/12 inch.

When the screw turns two complete revolutions, it travels a distance equal to the screw pitch. The spacing between neighboring threads is specified as the screw pitch. Because there is just one thread in this scenario, the pitch is 1/12 inch.

When the screw completes two complete rotations, it advances into the nut a distance equal to the screw pitch, which is 1/12 inch. Since the answer choices are given in fractions, we can simplify 1/12 to 1/6 by dividing both the numerator and denominator by 2. Hence, the correct answer is d) 1/6 inch.

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A singular point occurs when the coefficient of the highest derivative term, in this case y'', becomes zero. At x=1, the coefficient (x²-16)³(x-1) becomes 0, making x=1 a singular point for the given equation.

To determine if x=1 is a singular point for the equation (x²-16)³(x-1)y'' - 2xy' y = 0, we can examine the coefficients of the equation.

In more detail, a singular point in a differential equation is a point where the coefficients of the highest derivative terms are either undefined or equal to zero. For our equation, the highest derivative term is y'' and its coefficient is (x²-16)³(x-1). When x=1, this coefficient becomes (1²-16)³(1-1) = (1-16)³(0) = (-15)³(0) = 0.

Since the coefficient is equal to zero at x=1, it confirms that x=1 is indeed a singular point for the given equation.

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Step-by-step explanation:

Essentially Zero probability  ......   VERY unlikely that it will land on its edge.

18, 21, 24, 27, 30 can be gotten when 6, 7, 8, 9, 10 are multiplied three times.

Sequence is  an ordered list of numbers that often follow a specific pattern or rule. Sequence is a list of things that are in order.

How to determine this

6, 7, 8, 9, 10 are related to 18, 21, 24, 27, 30

6 * 3 = 18

7 * 3 = 21

8 * 3 = 24

9 * 3 = 27

10 * 3 = 30

All of them followed the same sequence of being multiplied by 3.

6, 7, 8, 9, 10 when multiplied thrice will give 18, 21, 24, 27, 30.

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Answer:

(a) Yes, the statement is true. If the null hypothesi is true, we can retain the null. 

(b) Yes, the statement is true

Answer:

(a) Yes, the statement is true. If the null hypothesi is true, we can retain the null. 

(b) Yes, the statement is true

The value of P(E ∪ F) is  0.67 so that correct answer is option d.

Given value of the  P(E) = 39⁄100 , P(Fc ) = 53⁄100 , and P(F ∩ Ec ) = 7⁄25, to find P(E ∪ F),

we can use the formula of probability;

P(E ∪ F) = P(E) + P(F) - P(E ∩ F)

step 1:-First, we need to find P(F). Since we know P(Fc) = 53/100, we can find P(F) using, here P(Fc) denotes probability of not happening event F.

P(F) = 1 - P(Fc) = 1 - (53/100) = 47/100

step2:-Next, we need to find P(E ∩ F). We know P(F ∩ Ec) = 7/25. Since Ec is the complement of E, we can use the formula:

P(E ∩ F) = P(F) - P(F ∩ Ec) = (47/100) - (7/25)

To subtract the fractions, we need a common denominator. The least common multiple of 100 and 25 is 100, so we convert (7/25) to (28/100):

P(E ∩ F) = (47/100) - (28/100) = 19/100

step3:-Now we can find P(E ∪ F) using the formula:

P(E ∪ F) = P(E) + P(F) - P(E ∩ F) = (39/100) + (47/100) - (19/100)

P(E ∪ F) = (39 + 47 - 19) / 100 = 67/100

So, the answer is d) 0.67.

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an+1 ≤ √(2(n+1)) + 1/2.

Since we have shown

To show that an = floor(√(2n) + 1/2), we need to prove two things:

an ≤ √(2n) + 1/2

an + 1 > √(2(n+1)) + 1/2

We will prove these statements by induction.

Base case: n = 1

a1 = 1 = floor(√(2*1) + 1/2) = floor(1.5)

The base case holds.

Induction hypothesis:

Assume that an = floor(√(2n) + 1/2) for some positive integer n.

Inductive step:

We need to show that an+1 = floor(√(2(n+1)) + 1/2) based on the induction hypothesis.

By definition of the sequence, a1 through an represent the first 1+2+...+n = n(n+1)/2 terms. Therefore, an+1 is the (n+1)th term.

The (n+1)th term is k if and only if 1+2+...+k-1 < n+1 ≤ 1+2+...+k.

Using the formula for the sum of the first k integers, we can simplify this condition to:

k(k-1)/2 < n+1 ≤ k(k+1)/2.

Multiplying both sides by 2 and rearranging, we get:

k^2 - k < 2n+2 ≤ k^2 + k.

Adding 1/4 to both sides, we get:

k^2 - k + 1/4 < 2n+2 + 1/4 ≤ k^2 + k + 1/4.

Taking the square root, we get:

k - 1/2 < √(2n+2) + 1/2 ≤ k + 1/2.

Now, we want to show that an+1 = k = floor(√(2(n+1)) + 1/2).

First, we will show that an+1 > √(2(n+1)) - 1/2.

Assume, for the sake of contradiction, that an+1 ≤ √(2(n+1)) - 1/2. Then:

k ≤ √(2(n+1)) - 1/2

k + 1/2 ≤ √(2(n+1))

(k + 1/2)^2 ≤ 2(n+1)

k^2 + k + 1/4 ≤ 2n + 2

This contradicts the fact that k is the smallest integer satisfying k^2 - k < 2n+2.

Therefore, an+1 > √(2(n+1)) - 1/2.

Next, we will show that an+1 ≤ √(2(n+1)) + 1/2.

Assume, for the sake of contradiction, that an+1 > √(2(n+1)) + 1/2. Then:

k > √(2(n+1)) + 1/2

k - 1/2 > √(2(n+1))

(k - 1/2)^2 > 2(n+1)

k^2 - k + 1/4 > 2n + 2

This contradicts the fact that k is the smallest integer satisfying 2n+2 ≤ k(k+1)/2.

Therefore, an+1 ≤ √(2(n+1)) + 1/2.

Since we have shown

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Answer:

Domain: (−∞,∞),{x|x∈R}(-∞,∞),{x|x∈ℝ}Range: (−∞,∞),{y|y∈R}, There are no vertical or horizontal asymptotes and x is 6.11

Step-by-step explanation:

Unsure on what Im solving for but here’s a couple different possibilities

In the circle on the left bottom, 4

the circle on the right bottom, 5

the square on the right, -15

Answer:

4/5

Step-by-step explanation:

The sin of B is equal to opposite/hypotenuse

So, the equation would be 4/5 because 4 is equal to the opposite side length of B and 5 is equal to the hypotenuse of the triangle.

So, the answer would be 4/5

Answer:

a. The total loss from buying and selling 300 shares at these prices can be calculated as follows:

Total cost of buying the stock = 300 shares x $41/share = $12,300

Total proceeds from selling the stock = 300 shares x $24.98/share = $7,494

Loss = Total cost - Total proceeds = $12,300 - $7,494 = $4,806

Therefore, the loss from buying and selling 300 shares of General Motors stock at these prices is $4,806.

b. To express the loss as a percent of the purchase price, we can use the following formula:

Loss percentage = (Loss / Total cost) x 100%

Substituting the values we found, we get:

Loss percentage = ($4,806 / $12,300) x 100% = 39.2%

Rounded to the nearest tenth of a percent, the loss percentage is 39.2%.

Answer:

e) ab/(a - b)

Step-by-step explanation:

1/x + 1/a = 1/b

now we need to transform this so that we get x = ...

first, let's multiply both sides by x

1 + x/a = x/b

now we subtract x/a from both sides

1 = x/b - x/a

we multiply both sides by a

a = ax/b - x

we multiply both sides by b

ab = ax - bx = x(a - b)

now we divide both sides by (a - b)

ab/(a - b) = x

Considering all Martha's instructions for making a cardboard cylinder, the resultant cardboard cylinder is present in above figure.

In mathematics, a cylinder is a three dimensional object with two parallel bases connected at fixed points by curved edges. The distance between the two bases is called the vertical distance and "h" indicates the height. The distance between two circles is called the radius of the cylinder and is defined by "r". It is combination of 2 circles + 1 rectangle :

We have provide the instructions follows by Martha to make a cardboard cylinder and we have to make same cylinder or like as Martha's.

After following the all these necessary instructions for cylinder construction, the resultant cardboard cylinder in present in figure.

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Complete question:

Martha must make a cardboard cylinder, she herself cuts out the bases, cuts the lateral face along a generatrix and extends it on the sheet; make a cylinder like Martha's.

The 95%-confidence interval is:

Confidence Interval = [67.0 - 2.76, 67.0 + 2.76] ≈ [64.24, 69.76]

a) The estimate of the height can be found by taking the average of the measurements:

Height Estimate = (67+69+65+68+66)/5 = 67.0

Therefore, the estimated height of the building is 67 meters.

b) The 99%-confidence interval can be found using the formula:

Confidence Interval = [Height Estimate - Margin of Error, Height Estimate + Margin of Error]

where the Margin of Error is given by:

Margin of Error = Z * (Standard Deviation / sqrt(n))

Here, n = 5 (number of measurements), Z is the value from the standard normal distribution that corresponds to the 99% confidence level, and the standard deviation is given as 4.

Using a table or calculator, we find that Z = 2.576.

Plugging in the values, we get:

Margin of Error = 2.576 * (4 / sqrt(5)) ≈ 4.14

Therefore, the 99%-confidence interval is:

Confidence Interval = [67.0 - 4.14, 67.0 + 4.14] ≈ [62.86, 71.14]

c) To estimate the standard deviation, we can use the sample standard deviation formula:

Sample Standard Deviation = sqrt(1/(n-1) * Sum((Xi - Xbar)^2))

where Xbar is the sample mean, Xi are the individual measurements, and n is the sample size.

Plugging in the values, we get:

Xbar = (67+69+65+68+66)/5 = 67.0

Sample Standard Deviation = sqrt(1/(5-1) * ((67-67)^2 + (69-67)^2 + (65-67)^2 + (68-67)^2 + (66-67)^2)) ≈ 1.58

Therefore, the estimated standard deviation for the measurement error is 1.58.

d) To find the 95%-confidence interval when the standard deviation is unknown, we can use the t-distribution with n-1 degrees of freedom. The formula for the confidence interval is:

Confidence Interval = [Height Estimate - Margin of Error, Height Estimate + Margin of Error]

where the Margin of Error is given by:

Margin of Error = t * (Sample Standard Deviation / sqrt(n))

Here, n = 5 (number of measurements), t is the value from the t-distribution that corresponds to the 95% confidence level and 4 degrees of freedom (n-1), and the sample standard deviation is 1.58 (calculated in part c).

Using a table or calculator, we find that t = 2.776.

Plugging in the values, we get:

Margin of Error = 2.776 * (1.58 / sqrt(5)) ≈ 2.76

Therefore, the 95%-confidence interval is:

Confidence Interval = [67.0 - 2.76, 67.0 + 2.76] ≈ [64.24, 69.76]

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Answer: D. The two rotations map the same image since 350° is a full rotation and 180° + 360° = 540°.

S = {1 - 2x², 1 + 3x - x², 1 + 2x + x³} is linearly independent in P3(R).

To determine if the set S is linearly independent in P3(R), we can use the linear combination method. We set a linear combination of the vectors in S equal to the zero vector:

c1(1 - 2x²) + c2(1 + 3x - x²) + c3(1 + 2x + x³) = 0

Now, we equate the coefficients of like terms:

c1 + c2 + c3 = 0
3c2 + 2c3 = 0
-2c1 - c2 + c3 = 0

This system of linear equations has only the trivial solution, where c1 = c2 = c3 = 0, which implies that the set S is linearly independent in P3(R).

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complete question:

Consider S = {1 − 2x^2, 1 + 3x − x^2, 1 + 2x + x3} ⊆ P3(R)  is S linearly independent in P3(R) ?

The length of side x in simplest radical form with a rational denominator is[tex]3\sqrt{4.5}[/tex].

An isosceles triangle is a triangle with any two sides that are the same length and angles on opposite sides that are the same size.

The right triangle is isosceles, which indicates that its two legs are the same length. This length should be called "y".

Using the Pythagorean theorem, we know that:

[tex]y^{2}+y^{2}=9^{2}[/tex]

Simplifying this equation:

[tex]2y^{2}=81[/tex]

Dividing both sides by 2:

[tex]y^{2}[/tex] = 40.5

Taking the square root of both sides:

y = [tex]\sqrt{40.5}[/tex]

We can simplify this expression by factoring out a perfect square:

y = [tex]\sqrt{4.5*9}[/tex]

y = [tex]3\sqrt{4.5}[/tex]

Since we know that x has the same length as y, the length of x is also:

x = [tex]3\sqrt{4.5}[/tex]

Therefore, the length of side x in simplest radical form with a rational denominator is[tex]3\sqrt{4.5}[/tex].

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Answer: Paul's front wheel completed 21,147 full revolutions.

Step-by-step explanation:

The distance traveled by the bike is equal to the circumference of the front wheel times the number of revolutions made by the wheel. The circumference C of a circle is given by the formula C = 2πr, where r is the radius of the circle.

In this case, the radius of the front wheel is 56 cm, so its circumference is:

C = 2πr = 2π(56 cm) ≈ 351.86 cm

To convert the distance traveled by Paul from kilometers to centimeters, we multiply by 100,000:

distance = 75.1 km = 75,100,000 cm

The number of full revolutions N made by the front wheel is therefore:

N = distance / C = 75,100,000 cm / 351.86 cm ≈ 213,470.2

However, we need to round down to the nearest integer since the wheel cannot complete a fractional revolution. Therefore:

N = 21,147

Therefore, Paul's front wheel completed 21,147 full revolutions.

Answer:

93.7 meters

Step-by-step explanation:

We can use similar triangles to solve this problem. Similar triangles are triangles that have the same shape but possibly different sizes. They have proportional sides.

Let's denote the height of the tower as 'h'. According to the given information, the height of the pole is 2.7m and it casts a shadow that is 1.27m long. Similarly, the tower casts a shadow that is 44.5m long.

We can set up a proportion using the heights and shadows of the pole and the tower:

height of pole / length of shadow of pole = height of tower / length of shadow of tower

Plugging in the values we have:

2.7 / 1.27 = h / 44.5

Now we can cross-multiply and solve for 'h':

2.7 * 44.5 = 1.27 * h

119.15 = 1.27h

Dividing both sides by 1.27:

h = 119.15 / 1.27

h ≈ 93.7

So, the height of the tower is approximately 93.7 meters, rounded to the nearest meter.