16. The ground area of a piece of land of a supermarket used for parking cars is 25(x² - 8x + 16) square metres.
(i) If the area of a parking lot for a vehicle is (x-4)² square metres, how many cars can be parked there?
(ii) If 4 units of the parking lot have been booked by the supermarket, how many parking lots are left? ​

a' = {13, 16, 17}

To find a', we need to identify the elements in u that are not present in a. Looking at the elements in u and comparing them with the elements in a, we can see that the elements 13, 16, and 17 are present in u but not in a.

Therefore, a' consists of these three elements: {13, 16, 17}. These elements are the elements in u that are not included in a.

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The cumulative distribution function (CDF) of the random variable X with the given probability density function (PDF) is F(x) = [tex]x^{4}[/tex], 0 <= x <= 1.

the probability that X is less than or equal to 0.5 is 0.0625.

The PDF, f(x) = 4[tex]x^{3}[/tex], is defined on the interval [0, 1]. To find the cumulative distribution function (CDF), we integrate the PDF from 0 to x.

∫[0, x] f(t) dt = ∫[0, x] 4[tex]t^{3}[/tex] dt = [tex]t^{4}[/tex] | [0, x] = [tex]x^{4}[/tex] - 0 = [tex]x^{4}[/tex]

So, the CDF of X is F(x) = [tex]x^{4}[/tex] for 0 <= x <= 1.

The CDF gives the probability that X takes on a value less than or equal to x. In this case, it means that F(x) = [tex]x^{4}[/tex] represents the probability that X is less than or equal to x.

For example, if we want to find the probability that X is less than or equal to 0.5, we substitute x = 0.5 into the CDF: F(0.5) = [tex]0.5^{4}[/tex] = 0.0625

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The correct statement are:

It is positive for all smooth paths C.

It is positive for all closed smooth paths C

The integral of F.dr over a smooth path C represents the circulation or line integral of the vector field F along the path C.

Since f is positive everywhere, Vf (the vector field derived from f) will also be a positive vector field.

"It is positive for all smooth paths C" is true because the line integral of a positive vector field over a smooth path will always be positive.

"It is positive for all closed smooth paths C" is also true because if the path C is closed, the line integral will be positive due to the positivity of the vector field and the fact that the path encloses a region where f is positive.

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Answer:

Find a unit vector that is parallel to the line tangent to the parabola y = x2 at the point (2, 4).

Summary:

The unit vector that is parallel to the line tangent to the parabola y = x2 at the point (2, 4) is ±(i + 4j)/√17.

Step-by-step explanation:

Find a unit vector that is parallel to the line tangent to the parabola y = x2 at the point (2, 4).

Solution:

Given parabola y = x2

Point (2, 4)

The slope of the tangent line to the parabola at (2,4) can be written as

(dy/dx) at (2,4) = 2x at (2,4) =4

So, any line parallel to the tangent line has slope ‘4’

Let us assume the unit vector is ±(i + 4j)

The length of the vector is √(12 + 42) = √17

So, the required unit vectors are ±(i + 4j)/√17

All the values are,

a)  lim x → 3 [ 2 f (x) - g (x)] = 18

b)  lim x → 3 [ 2 g (x) ]² = 16

c)  lim x → 3 [ ∛ f (x) / g (x) ] +  lim x → 3 [ 4 h (x) / x + 7 ] = - 1

We have to given that;

Limits are,

lim x → 3 f (x) = 8

lim x → 3 g (x) = - 2

lim x → 3 h (x) = 0

Now, We can simplify all the limits as;

1) lim x → 3 [ 2 f (x) - g (x)]

⇒  lim x → 3 [ 2 f (x)] -  lim x → 3 [ g (x) ]

⇒ 2  lim x → 3 [  f (x) ] - (- 2)

⇒ 2 × 8 + 2

⇒ 16 + 2

⇒ 18

2)  lim x → 3 [ 2 g (x) ]²

⇒ 4 [ lim x → 3  g (x) ]²

⇒ 4 × (- 2)²

⇒ 4 × 4

⇒ 16

3)  lim x → 3 [ ∛ f (x) / g (x) ] +  lim x → 3 [ 4 h (x) / x + 7 ]

⇒ ∛8 / (- 2) + 4 × 0 / (3 + 7)

⇒ - 2/2 + 0

⇒ - 1

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The mean of the underlying distribution for the chi-square is equal to the degrees of freedom (df) for that particular test.

The chi-square distribution is characterized by its degrees of freedom, which determine the shape of the distribution. The mean of the chi-square distribution is dependent on the degrees of freedom. The degrees of freedom represent the number of independent pieces of information used to estimate a parameter. In the case of the chi-square test, it is the number of categories or cells in a contingency table minus 1.

Therefore, the mean of the chi-square distribution is equal to the degrees of freedom for that specific test. Option (c) is the correct choice. The mean of the underlying distribution for the chi-square is not always 1.00 if the null is true (option a) or always 0.00 if the null is true (option b). Additionally, the mean cannot be determined without a critical values table (option d) since it is directly related to the degrees of freedom.

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The equation P = k * x * y * z expresses the direct proportionality between the variable P and the product of x, y, and z, with k as the constant of proportionality. Similarly, the equation S = k * (O^2 * v^2) / c^3

To express the proportionality relationship between P and the product of x, y, and z, we use the constant of proportionality, k, and write the equation as P = k * x * y * z. This means that P is directly proportional to the product of x, y, and z, and as the values of x, y, and z increase or decrease, P will increase or decrease proportionally.

For the second statement, we have S being proportional to the product of the squares of O and v, and inversely proportional to the cube of c. We can express this relationship using the constant of proportionality, k, as S = k * (O^2 * v^2) / c^3. The square of O and v is multiplied, while the cube of c is in the denominator, representing the inverse proportionality. As O, v, and c change, S will change accordingly, following the specified relationship.

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The height of the pyramid is approximately 12.5 inches.It's important to note that the result is rounded to two decimal places for simplicity, so the actual value may be slightly different.

To find the height of the pyramid, we can use the formula for the volume of a pyramid:

Volume = (1/3) * base area * height

Given that the volume of the pyramid is 1082.54 cubic inches and the base area is 259.81 square inches, we can substitute these values into the formula:

1082.54 = (1/3) * 259.81 * height

To solve for the height, we can multiply both sides of the equation by 3 and divide by the base area:

height = (1082.54 * 3) / 259.81

Calculating the right-hand side:

height ≈ 12.5 inches.

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Answer:

Step-by-step explanation:

You want the solutions to the triangles shown using the law of cosines and the law of sines.

The law of cosines tells you the relation between two sides and the angle between them:

  c² = a² +b² -2ab·cos(C)

Solving for the angle gives ...

  C = arccos((a² +b² -c²)/(2ab))

The law of sines tells you side lengths are proportional to the sines of their opposite angles.

  a/sin(A) = b/sin(B) = c/sin(C)

Knowing an opposite pair of side and angle, we can solve for the other sides or angles by rearranging:

  b = (a/sin(A))·sin(B)

  B = arcsin(b·sin(A)/a)

The sum of angles in a triangle is 180°, so we can always find the third angle once we know two of them.

We like to start by finding the largest angle, the one opposite the longest side. Here, that is ...

  R = arccos((6² +11² -13²)/(2·6·11)) ≈ 95.2°

The law of sines tells us another ange:

  Q = arcsin(11·sin(95.2°)/13) ≈ 57.4°

  P = 180° -95.2° -57.4° = 27.4°

The solution is (P, Q, R) = (27.4°, 57.4°, 95.2°).

Before we can use the law of sines, we need a side-angle pair. The only given side is opposite a missing angle, so we need to find that angle first.

  D = 180° -113° -25° = 42°

Then the other sides can be found.

  CD = sin(113°)·9/sin(42°) ≈ 12.4

  ED = sin(25°)·9/sin(42°) ≈ 5.7

The solution is (CD, ED, D) = (12.4, 5.7, 42°).

__

Additional comment

Solving the law of cosines formula for the missing side gives ...

  c = √(a² +b² -2ab·cos(C))

As you can see in the first attachment, these formulas are easily evaluated in one step using a suitable calculator. Intermediate values should always be preserved at full precision. Rounding should only be done on the final answers.

The last two attachments show an online triangle solver's solution to these problems. Some calculators have a triangle solver app built in. Stand-alone solver apps are also available.

<95141404393>

In converting binary numbers to decimal form, the binary numbers 111.111_2, 1000.0101_2, and 10101.011_2 can be converted to decimal as follows: 7.875, 8.3125, and 21.375, respectively.

To convert a binary number to decimal, each digit in the binary number is multiplied by the corresponding power of 2 and then summed. In the given binary numbers, the digits before the decimal point represent the whole number part, while the digits after the decimal point represent the fractional part.

For part D, the binary number 111.111_2 has three digits before the decimal point and three digits after. Starting from the left, the decimal equivalent can be calculated as (1 * 2^2) + (1 * 2^1) + (1 * 2^0) + (1 * 2^-1) + (1 * 2^-2) + (1 * 2^-3) = 7.875.

For part E, the binary number 1000.0101_2 has four digits before the decimal point and four digits after. Calculating the decimal equivalent gives (1 * 2^3) + (0 * 2^2) + (0 * 2^1) + (0 * 2^0) + (0 * 2^-1) + (1 * 2^-2) + (0 * 2^-3) + (1 * 2^-4) = 8.3125.

For part F, the binary number 10101.011_2 has five digits before the decimal point and three digits after. The decimal equivalent is (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0) + (0 * 2^-1) + (1 * 2^-2) + (1 * 2^-3) = 21.375.

By following the process of multiplying each digit by the corresponding power of 2 and summing the results, we can convert binary numbers to decimal form.

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Answer:

beta increases

Step-by-step explanation:

To find the conservative vector field F, we need to take the gradient of f(x, y, z):

F = ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Let's calculate the partial derivatives:

∂f/∂x = 6z/x

∂f/∂y = 7

∂f/∂z = 6ln(x)

Therefore, the conservative vector field F is:

F = (6z/x)i + 7j + 6ln(x)k

Now, let's evaluate the line integral of F over the circle (x-2)^2 + y^2 = 1 in the clockwise direction.

To evaluate the line integral, we need to parameterize the circle. Let's use the parameterization:

x = 2 + cos(t)

y = sin(t)

z = 0

where t ranges from 0 to 2π.

The differential of the parameterization is given by:

dr = (-sin(t)dt)i + (cos(t)dt)j + 0k

Now, we can calculate the line integral:

∫CF⋅dr = ∫[0 to 2π] (F⋅dr)

= ∫[0 to 2π] [(6z/x)i + 7j + 6ln(x)k]⋅[(-sin(t)dt)i + (cos(t)dt)j]

= ∫[0 to 2π] [-6zsin(t)/x + 7cos(t) + 6ln(x)cos(t)] dt

Note that since z = 0 and x = 2 + cos(t), we can simplify the integral further:

∫CF⋅dr = ∫[0 to 2π] [7cos(t)] dt

Integrating the cosine function over the interval [0, 2π] gives:

∫CF⋅dr = [7sin(t)] [from 0 to 2π]

= 7[sin(2π) - sin(0)]

= 7[0 - 0]

= 0

Therefore, the line integral of F over the circle (x-2)^2 + y^2 = 1 in the clockwise direction is 0.

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Answer:

90, 37

Step-by-step explanation:

we can say AC is diameter since the dot is center and p being angle subtended by arc AC, is 90° as angle at semicircle is 90.

sum of all sides is 360 so q is 37°

Based on the profit model provided, there are two break-even points for the product "sleeping bags." The x-coordinate of either one of these break-even points should be provided, rounded to three decimal places.

To determine the break-even points, we need to identify the x-coordinate where the profit is equal to zero. The break-even point represents the level of sales at which the company neither makes a profit nor incurs a loss.

The profit model should provide the necessary information to calculate the break-even points. However, the profit model or any specific details related to it were not provided in the question. Without the profit model or additional information, it is not possible to calculate the break-even points for the sleeping bags or provide the x-coordinate of either break-even point.

To determine the break-even points accurately, it is essential to have information such as fixed costs, variable costs, selling price per unit, and any other relevant factors that impact the profit of the sleeping bags. With this information, it is possible to calculate the break-even points using mathematical formulas and determine the corresponding x-coordinates.

In summary, without the profit model or any additional information, it is not possible to provide the x-coordinate of either break-even point for the sleeping bags.

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Complete Question:

Use the additional information and the profit model above to answer this question. there are two break-even points. give the -coordinate of either one. round to 3 decimal places. sleeping bags.

The angle between the vectors is 125°.

To find the angle between the vectors, we first need to calculate their dot product. Using the formula,

62 · −95 = (62)(−95)cosθ

we get -5890cosθ.

Next, we need to calculate the magnitude of each vector.

|62| = [tex]\sqrt{(62^{2})[/tex]= 62

|−95| = [tex]\sqrt{(95^{2})[/tex] = 95

Using the formula,

cosθ = (62 · −95) / (|62| · |−95|)

we get cosθ = -62/95.

Taking the inverse cosine,

θ = cos⁻¹(-62/95)

Using a calculator, we get θ ≈ 124.9°.

Rounding to the nearest degree, the angle between the vectors is 125°.

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Tom earned $1,370 after mowing 45 lawns.

The equation describing the total cost (C) of mowing for (l) lawns is:

C = 30l + 20

In this equation, the variable we represents the number of lawns mowed, and C represents the total cost.

The term 30l represents the cost of mowing each lawn, and the flat fee of $20 is added to it.

To calculate the amount of money Tom earned after mowing 45 lawns, we can substitute l = 45 into the equation:

C = 30(45) + 20

C = 1350 + 20

C = 1370

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In a comparative study to determine differences in mean scores between a group of teachers, engineers, and nurses, an appropriate statistical analysis would be analysis of variance (ANOVA) followed by post hoc tests.

Analysis of variance (ANOVA) is a suitable statistical analysis for comparing mean scores across multiple groups. In this case, it allows us to examine the differences in mean scores between the group of teachers, engineers, and nurses. ANOVA assesses whether there are significant differences in means by analyzing the variation between groups and within groups. If the ANOVA results indicate a significant difference, post hoc tests can be conducted to determine which specific groups differ from each other.

Commonly used post hoc tests for ANOVA include Tukey's honestly significant difference (HSD), Bonferroni correction, or Dunnett's test. These tests help identify the specific pairs of groups that have significantly different mean scores. By conducting post hoc tests, we can gain a more detailed understanding of the nature and magnitude of the differences between teachers, engineers, and nurses in terms of their mean scores.

In conclusion, to determine differences in mean scores between a group of teachers, engineers, and nurses, an appropriate statistical analysis would involve conducting an analysis of variance (ANOVA) followed by post hoc tests to compare specific group differences.

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Answer:

(x - 10)² + (y + 4)² = 121

Step-by-step explanation:

the equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r is the radius

here (h, k ) = (10, - 4 ) and r = 11 , then

(x - 10)² + (y - (- 4) )² = 11² , that is

(x - 10)² + (y + 4)² = 121

The exact value of cos(θ) = √(21/25) = √21/5.

We can use the trigonometric identity cos(-θ) = cos(θ) to find the value of cos(-θ) using the given information.

Given that sin(θ) = 2/5, we can use the Pythagorean identity[tex]sin^2(\theta) + cos^2(\theta) = 1[/tex] to find the value of cos(θ).

[tex]sin^2(\theta) + cos^2(\theta ) = 1[/tex]

[tex](2/5)^2 + cos^2(\theta) = 1[/tex]

[tex]4/25 + cos^2(\theta) = 1[/tex]

[tex]cos^2(\theta) = 1 - 4/25[/tex]

[tex]cos^2(\theta) = 25/25 - 4/25[/tex]

[tex]cos^2(\theta) = 21/25[/tex]

Taking the square root of both sides, we find:

cos(θ) = ±√(21/25)

Since 0 < θ < 90, we know that cos(θ) is positive.

Therefore, cos(θ) = √(21/25) = √21/5.

Using the identity cos(-θ) = cos(θ), we can conclude that cos(-θ) = √21/5.

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Answer:b and a

Step-by-step explanation:

The 90% confidence interval for the population proportion of defective disks is approximately 0.022 to 0.077

To construct a confidence interval for the population proportion of defective disks, we can use the formula for a confidence interval for a proportion:

Confidence Interval = Sample Proportion ± Margin of Error

The sample proportion (p-hat) is calculated by dividing the number of defective disks (16) by the total number of disks in the sample (322):

Sample Proportion (p-hat) = 16/322 ≈ 0.0497

The margin of error can be calculated using the formula:

Margin of Error = Critical Value * Standard Error

The critical value depends on the desired confidence level. For a 90% confidence level, we can use a Z-score corresponding to a 5% significance level in each tail. This corresponds to a critical value of approximately 1.645.

The standard error is calculated as the square root of [(p-hat * (1 - p-hat)) / n], where n is the sample size.

Standard Error = √[(0.0497 * (1 - 0.0497)) / 322] ≈ 0.0168

Now we can calculate the margin of error:

Margin of Error = 1.645 * 0.0168 ≈ 0.0277

Finally, we can construct the confidence interval:

Confidence Interval = 0.0497 ± 0.0277

Lower Limit = 0.0497 - 0.0277 ≈ 0.0220

Upper Limit = 0.0497 + 0.0277 ≈ 0.0774

Therefore, the 90% confidence interval for the population proportion of defective disks is approximately 0.022 to 0.077.

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The required rate of return for the Global Investment Fund will be = 7.00% + 0.9375 × (13.25% - 7.00%) = 10.50%

To calculate the required rate of return for the Global Investment Fund, we need to consider the risk of the individual stocks held in the portfolio. The required rate of return is influenced by the market's required rate of return and the risk-free rate, as well as the systematic risk of each stock, measured by its beta.

Given the information provided, we have the following investments and betas for each stock:

Stock A: Investment $200,000, Beta 1.50

Stock B: Investment $300,000, Beta -0.50

Stock C: Investment $500,000, Beta 1.25

Stock D: Investment $1,000,000, Beta 0.75

To calculate the portfolio beta, we need to weight the individual betas by their respective investments. In this case, the portfolio beta is 0.9375.

Using the Capital Asset Pricing Model (CAPM), we can calculate the required rate of return for the portfolio:

Required Rate of Return = Risk-Free Rate + Portfolio Beta * (Market's Required Rate of Return - Risk-Free Rate)

Plugging in the values, we get:

Required Rate of Return = 7.00% + 0.9375 × (13.25% - 7.00%) = 10.50%

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The given problem involves a projectile launched at an initial speed of 500 m/s and an angle of elevation of 30 degrees.

By applying the equations of projectile motion, we can determine important characteristics of the projectile's trajectory. The range, which is the horizontal distance covered by the projectile, is approximately 8984.7 meters.

The maximum height reached by the projectile is approximately 637.76 meters. The speed at impact refers to the velocity magnitude when the projectile hits the ground. It can be calculated by decomposing the initial velocity into horizontal and vertical components.

The horizontal component remains constant at around 433.01 m/s, while the vertical component is determined by the time taken to reach the ground.

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The continuous random variables in the given options are:

C. X is the average number of petals per daisy computed from all the daisies found in a randomly chosen grassy area 1 square meter in size.

D. X is the stem length in centimeters of a randomly chosen daisy.

The continuous random variables are those that can take on any value within a range or interval. Let's analyze each option:

A. X is the number of petals on a randomly chosen daisy.

This variable is discrete because the number of petals can only take on specific whole number values. For example, a daisy can have 5 petals, 6 petals, 7 petals, etc. It cannot have fractional or continuous values.

B. X is the number of daisies found in a randomly chosen grassy area 1 square meter in size.

This variable is discrete because the number of daisies can only be a whole number. You can count the number of daisies in the area, and it will give you a specific count, not a continuous value.

C. X is the average number of petals per daisy computed from all the daisies found in a randomly chosen grassy area 1 square meter in size.

This variable is continuous because the average number of petals per daisy can take on any real number value within a certain range. The average can be a fractional or decimal value, allowing for a continuous range of possibilities.

D. X is the stem length in centimeters of a randomly chosen daisy.

This variable is continuous because the stem length can take on any real number value within a certain range. The length can be fractional or decimal, allowing for a continuous range of possibilities.

Therefore, the continuous random variables in the given options are:

C. X is the average number of petals per daisy computed from all the daisies found in a randomly chosen grassy area 1 square meter in size.

D. X is the stem length in centimeters of a randomly chosen daisy.

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The function f(x) = x² - 3  has a unique minimum point, and when evaluating the function at a value a, we have f(a)=a²−3.

To verify if the function f(x) = x² - 3 has an inverse function, we need to check if it is one-to-one. Let's analyze the given options to determine the correct statement.

Based on the given options:

The domain of f is all real numbers: This is true since there are no restrictions on the values of x for the function f(x) = x² - 3.

f has exactly one maximum: False, as the function f(x) = x² - 3 does not have a maximum value. It is an upward-opening parabola with the vertex at the point (0, -3).

The range of f is all real numbers: False, as the range of the function f(x) = x² - 3 is limited to y ≥ -3 (all values greater than or equal to -3).

f has exactly one minimum: True, the function f(x) = x² - 3 has a minimum value at the vertex (0, -3).

f is one-to-one: False, since the function f(x) = x² - 3 fails the horizontal line test, indicating that it is not one-to-one.

Based on the analysis above, the correct statement is: f has exactly one minimum.

To find f(a), we substitute the given real number a into the function f(x) = x² - 3:

f(a) = a² - 3

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The next number in the series is -85.

To determine the pattern in the series, we observe that each number is obtained by subtracting a decreasing sequence of numbers from the previous number.

298 - 89 = 209

209 - 80 = 129

129 - 71 = 58

58 - 62 = -4

The sequence of subtracted numbers is decreasing by 9 each time. So, we continue this pattern and subtract 53 from -4 to obtain the next number:

-4 - 53 = -57

Thus, the next number in the series is -85.

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The calculated area of the largest square is 169 square units

From the question, we have the following parameters that can be used in our computation:

The shapes (see attachment)

We have

Area 3 = 25

Perimeters 2 = 48

This means that

Side length 3 = 5

Side length 2 = 12

The square 1 is the largest square

So, we have

Area of square 1 = Side length 1² + Side length 3²

So, we have

Area of square 1 = 5² + 12²

Evaluate

Area of square 1 = 169

Hence, the area of the square 1 is 169

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          A

         / \

       /   \

     /     \

  B-------C

Since B is the midpoint of AC, we know that AB = BC = 2.

Since C is the midpoint of AD, we know that AC = CD.

Using the Pythagorean Theorem in triangle ABC, we have:

AB^2 + BC^2 = AC^2

2^2 + 2^2 = AC^2

8 = AC^2

AC = 2√2

Since AC = CD, we have CD = 2√2.

Therefore, the length of CD is 2 units.

Answer: 104

Step-by-step explanation:

A percent divided by 100 becomes a decimal.

      40% / 100 = 0.4

Next, "of" ("de") means multiplication in mathematics.

      0.4 * 260 = 104