Evaluate the line integral c y3 ds,
c.x = t3, y = t, 0 ≤ t ≤ 3

The product is equivalent to $72.

A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.

We have a rate of $9 per hour for 8 hours.

Now, for 1 hour, the cost is $9. Therefore, for 8 hours, we can write -

x = 9 x 8

x = $72

Therefore, the product is equivalent to $72.

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The 75th derivative of y = cos(2x) can be found by dividing 75 by 4 to find the cycle it falls into due to the repeating pattern of cos and sin. Due to remainder 3 after division, the 75th derivative is equivalent to the 3rd derivative from the cycle, which is 8sin(2x), but scaled by 2 to the power of the number of completed cycles (18). This gives the final answer 2^18 * 8sin(2x).

The process of finding the 75th derivative of a function can seem daunting or time-consuming, but with functions like cosines and sines, we can rely on the cycling pattern of these functions to simplify the task. This will significantly decrease the amount of computational effort needed.

When you find the derivative of y = cos(2x), the function cycles in a pattern every four times. The first derivative of y = cos(2x) is -2sin(2x), the second derivative is -4cos(2x), the third derivative is 8sin(2x), and the fourth derivative is 16cos(2x). As you can see, we've returned to the original function (scaled by a factor of 16).

So to find the 75th derivative, divide 75 by 4 to get remainder of 3. This means the 75th derivative is the same as the 3rd derivative, but scaled by 2 to the power of the number of full cycles (which is 18). Hence, the 75th derivative of y = cos(2x) is 2^18 * 8sin(2x).

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The height of the triangular base is  155.46 m.

To find the height of the equilateral triangle base of the prism, we'll first determine the height of one of its constituent triangles.

The height ( h ) of an equilateral triangle is given by [tex]\( h = \sqrt{3}/2 \times \text{side length} \).[/tex]

Calculate the height of one triangle:

[tex]\[ h = \frac{\sqrt{3}}{2} \times 180 \, \text{m} = \frac{\sqrt{3} \times 180}{2} = \frac{180\sqrt{3}}{2} = 90\sqrt{3} \, \text{m} \][/tex]

Calculate the total height of the prism:

The height of the prism is given as 25 m.

Calculate the total height:

Total height = height of one triangle + height of the prism

[tex]\[ \text{Total height} = 90\sqrt{3} \, \text{m} + 25 \, \text{m} = 90\sqrt{3} + 25 \, \text{m} = 155.46 \, \text{m} \][/tex]

So, the height of the triangular base is  155.46 m

The temperature fall can be represented by −10. To find the average change per hour, divide −10 by 2. −10 divided by 2 is −5.

Final answer:

To find the most extreme 5% of normally distributed scores with a mean of 45 and a standard deviation of 6, we look for scores beyond the z-score cut-offs corresponding to 2.5% and 97.5%. The z-score around 1.96 gives us two scores: 56.76 on the high end and 33.24 on the low end, indicating that answer 'c' is correct.

Explanation:

To determine the most extreme 5% of the scores in a normally distributed population with a mean of 45 and a standard deviation of 6, we need to find the z-score that corresponds to the tails of the distribution where the cumulative area is either less than 2.5% or greater than 97.5% (since 5% is split into two tails of the distribution).

Using a standard normal distribution table or a calculator, the z-score that corresponds to the upper 97.5% is typically about 1.96 (or sometimes 1.645 for a one-tailed test). To find the actual scores corresponding to these z-scores, we use the formula for transforming a z-score into an actual value which is:
X = μ + (z • σ)

For the upper extreme:
X = 45 + (1.96 • 6) = 56.76

For the lower extreme:
X = 45 - (1.96 • 6) = 33.24

Thus, the most extreme 5% of the scores lie beyond scores of 56.76 and 33.24. Therefore, the correct answer is c. 56.76 and 33.24.

Answer:

(a) The density of flag is [tex]\dfrac{25}{27}[/tex] Pounds per square foot

(b) The weight of strip is [tex]\dfrac{20}{9}(15-h)\Delta h[/tex] Pounds

(c) The work is [tex]\dfrac{20}{9}(15-h)h\Delta h[/tex] foot-pounds

(d) The exact work by roof is 1250 foot-pounds

Step-by-step explanation:

(a) We are given a flag in the shape of a right triangle.

The total weight of flag is 250 pounds and Uniform density.

Base of the flag [tex]=\sqrt{b^2-a^2}[/tex]

                          [tex]=\sqrt{39^2-15^2}=36[/tex]

Area of the flag [tex]=\dfrac{1}{2}\times 36\times 15 = 270[/tex]

Weight of flag = 250 pounds

[tex]Density =\dfrac{Weight}{Area}=\dfrac{250}{270}=\dfrac{25}{27}[/tex]

Hence, The density of flag is [tex]\dfrac{25}{27}[/tex]

(b) Weight of strip which is h feet below the roof.

Length of strip [tex]=\dfrac{36}{15}(15-h)[/tex]

Width of strip [tex]\Delta h[/tex]

Weight = Density x area

            [tex]=\dfrac{25}{27}\times \dfrac{36}{15}(15-h)[/tex]

            [tex]=\dfrac{20}{9}(15-h)\Delta h[/tex]

Hence, The weight of strip is [tex]\dfrac{20}{9}(15-h)\Delta h[/tex]

(c) Work slice to move h feet above to the roof

work[tex]=Weight\times displacement[/tex]

       [tex]=\dfrac{20}{9}(15-h)\Delta h\times h[/tex]

       [tex]=\dfrac{20}{9}(15-h)h\Delta h[/tex]

Hence, The work is [tex]\dfrac{20}{9}(15-h)h\Delta h[/tex] foot-pounds

(d) Exact work on the roof by hanging flag

[tex]W=\int_0^{15}\dfrac{20}{9}(15-h)hd h[/tex]

[tex]W=\dfrac{20}{9}(\dfrac{15h^2}{2}-\dfrac{h^3}{3})|_0^{15}[/tex]

[tex]W=1250-0[/tex]

[tex]W=1250[/tex]

Hence, The exact work by roof is 1250 foot-pounds

Final answer:

The weight of the flag is distributed uniformly across its area, with the entire 250 pounds concentrated at the centroid point.

Explanation:

To find the centroid of a right triangle, we can use the following formulas:

[tex]\[ x_c = \frac{b}{3} \][/tex]

[tex]\[ y_c = \frac{a}{3} \][/tex]

Where:

[tex]- \( x_c \)[/tex] is the x-coordinate of the centroid.

[tex]- \( y_c \)[/tex] is the y-coordinate of the centroid.

[tex]- \( a \)[/tex] is the length of the side perpendicular to the base (height).

[tex]- \( b \)[/tex] is the length of the base.

Given \( a = 15 \) and \( b = 39 \), we can calculate the centroid as follows:

[tex]\[ x_c = \frac{39}{3} = 13 \][/tex]

[tex]\[ y_c = \frac{15}{3} = 5 \][/tex]

So, the centroid of the triangle is located at the point (13, 5).

Now, regarding the weight distribution, if the flag has uniform density, the center of mass (or centroid) would be the point where the total weight is concentrated. Since the total weight of the flag is 250 pounds, and the centroid is the point of concentration of mass, the entire weight of the flag, 250 pounds, can be considered to act at the centroid.

Therefore, the weight of the flag is distributed uniformly across its area, with the entire 250 pounds concentrated at the centroid point.

The pair of measurements that is not equivalent is 3 miles and 15,480 feet. All other pairs are equivalent when converted to the same units. This determines which measurements do not match in the same units.

To determine which pair of measurements is not equivalent, we need to convert them into the same units and compare them:

Thus, the pair of measurements that is not equivalent is 3 miles, 15,480 feet.

Answer: 8.04 meters

Step-by-step explanation:

length of first piece of yarn  = 5.89 meters

length of second piece of yarn = 2.147 meters

Total length of  of the two pieces of yarn = length of first piece of yarn + length of second piece of yarn = 5.89 meters + 2.147 meters = 8.037 meters

The rule apply for the addition and subtraction is :

The least precise number present after the decimal point determines the number of significant figures in the answer.

Thus the answer will be 8.04 meters as least precise number 5.89 contains two decimal places.