The illumination of a light source varies inversely with the square of
the distance from that source. At a distance of 120 meters away from
the source, the illumination is 20 lux. What is the formula for the
illumination, E, in lux, as a function of the distance, d, from the source,
in meters?

The expression: lim θ→π/2 sin(θ)/(4cos(2θ)) = sin(π/2)/(4cos(2π/2)) = 1/4.

The limit is 1/4.

What is trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

We can begin by directly substituting π/2 into the expression and see that we get an indeterminate form of 0/0:

lim θ→π/2 (1 - sin(θ))/(1 + cos(2θ)) = (1 - sin(π/2))/(1 + cos(2π/2)) = 0/0.

To apply l'Hospital's Rule, we take the derivative of the numerator and denominator with respect to θ:

lim θ→π/2 (1 - sin(θ))/(1 + cos(2θ)) = lim θ→π/2 (-cos(θ))/(−2sin(2θ))

Now we can directly substitute π/2 into the expression:

lim θ→π/2 (-cos(θ))/(−2sin(2θ)) = (-cos(π/2))/(−2sin(2π/2)) = -1/0,

which is another indeterminate form. We can apply l'Hospital's Rule again by taking the derivative of the numerator and denominator with respect to θ:

lim θ→π/2 (-cos(θ))/(−2sin(2θ)) = lim θ→π/2 sin(θ)/(4cos(2θ))

Now we can substitute π/2 into the expression:

lim θ→π/2 sin(θ)/(4cos(2θ)) = sin(π/2)/(4cos(2π/2)) = 1/4.

Therefore, the limit is 1/4.

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The main answer to your question is "interchanging". To find the inverse of a function, we interchange the numbers in each ordered pair of the function. This means that we switch the x and y values of each point in the function.

For example, if we have a function f(x) = 2x + 3, the ordered pairs would be (1,5), (2,7), (3,9), etc. To find the inverse function, we would switch the x and y values of each point to get ordered pairs such as (5,1), (7,2), (9,3), etc.

The explanation for why we interchange the numbers is that the inverse function "undoes" the original function. If we apply the original function to a number, the inverse function will take us back to the original number. By switching the x and y values, we make sure that the inverse function will undo the original function.

In conclusion, to find the inverse of a function, we interchange the numbers in each ordered pair of the function. This ensures that the inverse function will undo the original function.
Hi! I'm happy to help you with your question.

Main answer: The inverse of a function can be found by interchanging the numbers in each ordered pair of the function.

Explanation: When finding the inverse of a function, you are essentially swapping the input and output values in each ordered pair (x, y) to create a new ordered pair (y, x). This process is called interchanging the numbers in the ordered pair.

Conclusion: To find the inverse of a function, you need to interchange the numbers in each ordered pair of the function, which essentially swaps the input and output values.

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The number 0. 07129813 corrected to 2 decimal places is 0.07

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

0.07129813

Corrected to 2 decimal places means that we leave only two digits after the decimal points

using the above as a guide, we have the following:

0.07129813  = 0.07

Hence, the solutuion is 0.07

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The probability that sum of the two dice on a roll is 5 given that the blue die for that roll landed on 6 is 0.25 or 25%.

If we know that the blue die landed on 6, then we only need to consider the possible outcomes of the roll of the red die that would result in a sum of 5.

If red die shows 1, 2, 3 or 4, then the sum of the dice will be 6. There are a total of 6 possible outcomes for the roll of the red die (since it is a standard 6-sided die), but we can eliminate the outcomes 5 and 6 since they would result in a sum greater than 5.

So, out of the 4 possible outcomes for the roll of the red die that would result in a sum of 5, only one of them will occur if the blue die landed on 6. Therefore, the probability of getting a sum of 5 given that the blue die landed on 6 is:

1/4 = 0.25 or 25%

Hence the probability of the sum being 5 is 0.25 or 25%.

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A graph of the piecewise function is shown on the coordinate plane in the image attached below.

In Mathematics, a piecewise-defined function can be defined as a type of function that is defined by two (2) or more mathematical expressions over a specific domain.

Generally speaking, the domain of any piecewise-defined function simply refers to the union of all of its sub-domains. By critically observing the graph of the given piecewise-defined function, we can reasonably infer and logically deduce that it is decreasing over the interval x ≤ 1 and increasing over the interval x > 1.

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Answer:

[tex]x = \sqrt{ {17}^{2} - {15}^{2} } = \sqrt{289 - 225} = \sqrt{64} = 8[/tex]

So x = 8 = 8.0

Answer:

65 apples

Step-by-step explanation:

We Know

John has 8 boxes of apples.

Each box holds 10 apples.

If 5 of the boxes are full, and 3 of the boxes are half full, how many apples does John have?

Let's solve

5 boxes are full: 5 x 10 = 50 apples

3 boxes are half full = 3(1/2 · 10) = 15 apples

50 + 15 = 65 apples

So, John has 65 apples.

The probability that a single randomly selected value is greater than 203.4 is approximately 0.556.

To solve this problem, we need to use the properties of the normal distribution and probability.

First, we know that the population has a normal distribution with a mean of 208.5 and a standard deviation of 35.4. This means that the distribution of sample means will also be normal with a mean of 208.5 and a standard deviation of 35.4/sqrt(236), which is approximately 2.3.

Next, we want to find the probability that a single randomly selected value is greater than 203.4. To do this, we need to standardize the value using the formula:

z = (x - μ) / σ

where x is the value we want to find the probability for, μ is the population mean, and σ is the population standard deviation.

Plugging in the values we have:

z = (203.4 - 208.5) / 35.4 = -0.144

This means that the value of 203.4 is 0.144 standard deviations below the mean.

Using a standard normal distribution table or calculator, we can find the probability of a value being less than -0.144, which is the same as the probability of a value being greater than 0.144. This probability is approximately 0.556.

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Using integration by part, the value of the integration over the interval is given as [(4√3 π - π + 2ln(2) - 4)] / 6

In the given question, we have a function which is;

f(x) = √x tan⁻¹ √x dx over an interval of lower limit 1 and upper limit of 3

Applying integration by parts

[2/3x^3/2 arctan(√x) - ∫ x/3(x + 1) dx]^3_1

We can simplify this into;

[1/3(2x^3/2 arctan(√x) - x - 1 + ln |x + 1|)] ^3_1

Let's compute the boundaries

[(4√3 π - π + 2ln(2) - 4)] / 6

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Thus, option B, "The population mean is between 8 and 10," is the claim that is most supported by the interval.

Based on the given confidence interval of 6 to 14, we can say with 95% confidence that the true population mean falls between those values. Therefore, option B, "The population mean is between 8 and 10," is the claim that is most supported by the interval.

Option A, "The population mean is less than 15," and option E, "The population mean is more than 7," are also supported by the interval but are less specific than option B.

Option C, "The population mean is exactly 9," is not necessarily supported by the interval, as the true mean could be any value within the interval.

Option D, "The population mean is more than 17," is not supported by the interval, as the upper limit of the interval is only 14.

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Answer: C. k is equal to one fourth

Step-by-step explanation:

formula for parabola in vertex form

y= a(x-h)²+k     (h, k) is vertex here it is (0,0)

f(x)=ax²   another point we can plug in is (1,2)

2=a1

a= 2

so f(x)= 2x²

g(x)= f(kx)     plug in kx into f(x)

g(x) = 2(kx)²

g(x) =  2(k²)(x²)    plug in a point and find k (4,2)

2 = 2 k²4²

k²=1/16

k=1/4

C

Heart = 1

Water = 2

star = 4

Step-by-step explanation:

Heart x water = water, so heart is probably 1.

Water x water=star and water + water= star so water is 2. Star is 4

Answer:

heart= 1
tear= 2
star=4

leaf=2
diamond=3
circle=6

Step-by-step explanation:

if heart x tear equals tear one of those numbers has to be one.

so heart equals 1
tear + tear= star and tear x tear= star
only number that work are 2 since 2+2=4 and 2x2=4

leaf +leaf +leaf = circle
so plug in 2 for the leaves. 2+2+2=6
2x ____=6 plug in 3 so diamond is 3
3+3=6 so those are the correct choices.

The measure of the angles of the isosceles triangle is x = 36°

Given data ,

Let the measure of one angle of the isosceles triangle be = 108°

where , the isosceles triangle has three acute angles, meaning that the angles are less than 90°

So , let the measure of the unknown angle be x

And , x + x + 108 = 180

Subtracting 108 on both sides , we get

2x = 72

Divide by 2 on both sides , we get

x = 36°

Hence , the angle of isosceles triangle is x = 36°

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The complete question is attached below :

One angle of an isosceles triangle measures 108°. What measures are possible for the other two angles? Choose all that apply.

Answer:

x=80

Step-by-step explanation:

x+12 and x+48 must equal 180 so 60 is the answer

60+12=72

60+48=108

72+108=180

Answer: 60

Step-by-step explanation:

When you have 2 parrallel lines the angles will either be = or =180 in this case =180

so

A+B=180

x + 12 + x + 48 = 180

2x + 60 =180

2x = 120

x=60

Answer:

18.3

Step-by-step explanation:

In order to convert Fahrenheit to Celsius, you need to use the formula:

[tex]C=\frac{5}{9}(F-32)[/tex]

Since F = 65, we can plug it in:

[tex]C=\frac{5}{9}(65-32)[/tex]

[tex]C=\frac{5}{9}(33)[/tex]

[tex]C=\frac{165}{3}[/tex]

[tex]C=18.33333333333333333333333333[/tex]

Rounding to the nearest tenth, we get

[tex]C = 18.3[/tex]

In the given graph, the length of the segment "AB" is 6.32units.

We have to find length of segment "AB", which means we have to find the distance between the end-points "A" and "B",

From the graph, the end-points of the segment "AB" are :

A ⇒ (-2,4) and B ⇒ (4,2),

So, the length(distance) between these two points can be calculated by the formula : √((x₂-x₁)² + (y₂-y₁)²);

Considering (-2,4) as (x₁, y₁) and (4,2) as (x₂, y₂);

We get,

Length(distance) = √((4-(-2))² + (2-4)²);

Length = √(6² + (-2)²); = √(36 + 4);

Length = √40 ≈ 6.32 units.

Therefore, the length of the segment is 6.32 unis.

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Any number above 108 will be an upper outlier. So, there is no upper outlier.

Therefore, there is 1 outlier.

Mean  :

For a given distribution, the mean can be calculated by dividing the sum of all quantities with the number of quantities. Sometimes it referred to as average.

Median:

The median of a given data distribution is calculated as follows:

1. First arrange all the given data in ascending order.

2. If the number of data items is odd, then the centre data represents the median.

3. If the number of data items is even, then two numbers are in the central data. The average of those two data items is a median.

Mode

In the given data, the element which occurs number of times is called the mode of the data.

Step 2/14

a.

Consider the stem and leaf plot given in the text book that represents the money spent on gas in the last week by the selected seniors.

Compute the frequency of the given stem-and-leaf plot by counting its leaves. There are 27 leaves in the given plot.

Therefore, a total 27 students were polled.

Step 3/14

b.

Add the data represented in the plot and divide it by the frequency to compute the mean.

The sum of all data points is as follows,

1662

So, the mean of the given data set will be,

1662/27

=$61.56

Step 4/14

c.

The frequency of the given data set is an odd number (27).

So, the median will be the 14th data point.

From the given data set, 14th data point is $ 64.

Therefore, the median of the given data is. $64

Step 5/14

d.

The Mode of a given data set is the most occurring item.

From the given stem and leaf plot, the leaf 3 corresponding to the stem 5 is the most occurring data point (5 times).

Therefore, the mode of the given data is. $53

Step 6/14

e.

The Range is the difference between the greatest number and the least number in the given data set.

The lowest data point is 17 $ and the greatest data point is 84 $

Therefore, the range of the given data will be,

Range = 84 -17

=$67

Step 7/14

f.

The lower quartile of the given data set will be the 7th data point.

Therefore, Q1 = $54

The second quartile of the given data set will be equal to the median.

Therefore, Q2 = $64

The upper quartile of the given data set will be the 21st data point.

Therefore, Q3 = $75

Forth quartile of the given data set will be equal to the maximum value in the data set. Therefore, Q4 =-$84

Step 8/14

g.

53 $ is the lower quartile of the given data set. So, 25% of the numbers in the data set are at or below 53 $.

Therefore, 75% of the students spent $ 53 or more on gas.

Step 9/14

h.

Inter quartile range of the given data set will be,

Q3 -Q1

=$22

Step 10/14

i.

53 $ is the lower quartile and 75 $ is the upper quartile of the given data set. 50% of the numbers in the data set lie between lower quartile and upper quartile.

Therefore, 50% of the students spent from $ 53 to $ 75 on gas.

Step 11/14

j.

The boundary for the lower outliers will be,

53- 1.5 x 22

=20

Step 12/14

k.

The boundary for the upper outliers will be,

75 + 1.5 x 22

=108

Step 13/14

l.

Any number below 20 will be a lower outlier. So, $ 17 is a lower outlier.

Any number above 108 will be an upper outlier. So, there is no upper outlier.

Therefore, there is 1 outlier.

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Answer:

The answer to your question is $375

The selling price of the cell phone is $375.

Step-by-step explanation:

15% x 2500

100 = 375

I hope this helps and have a wonderful day!

The solution to the given proportion is x = 6

A proportion is an equation that sets two ratios equal to each other. For example, if there is one boy and three girls, the ratio may be written as: 1: 3 (for every one boy, there are three girls).

To determine if a connection is proportional, examine the ratios between the two variables. The connection is proportionate if the ratio is always the same. The connection is not proportional if the ratio changes.

To solve the proportion, we state:

x/3 = 6/3

x / 3 = 2

x = 2 x 3

x = 6

Thus, the solution to the proportion is x = 6

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Full Question:

Although part of your question is missing, you might be referring to this full question:

Solve the given proportion.

x/3 = 6/3

The probability of hitting the center target is 3.8%.

To find the probability of hitting the center target, we need to know the total area of the larger target and the area of the center target.

The area of the larger target is:

16 inches x 20 inches = 320 square inches

The area of the center target is:

4 inches x 3 inches = 12 square inches

So, the probability of hitting the center target is:

12 square inches / 320 square inches = 0.0375 or 3.75%

Rounded to the nearest tenth of a percent, the probability of hitting the center target is 3.8%.

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Answer:
Range: Men: 10, Women: 11
MAD: Men: 2.833333 (17/6), Women: 3

Step-by-step explanation:

The range of the men's scores is 75-65 (highest-lowest), which is 10.

The range of the woman's scores is 80-69, which is 11.

The MAD is the mean of the absolute difference between the terms and the mean, which is very painful to do, but in the sake of the problem, I will be doing.

Men's MAD:
The mean is [tex]\frac{65+68+70+72+73+75}{6}[/tex], which is 70.5.

Sigh.

Now, we find the difference of each term to the mean.

[tex]70.5-65= 5.5[/tex]

[tex]70.5-68=2.5[/tex]

[tex]70.5 - 70 = 0.5[/tex]

[tex]72-70.5=1.5[/tex]

[tex]73-70.5=2.5[/tex]

[tex]75-70.5=4.5[/tex]

wow. Now, we find the mean of these numbers.

[tex]\frac{5.5 + 2.5 + 0.5 + 1.5 + 2.5 + 4.5}{6}[/tex]= 17/6=2.8333333333 This is the MAD.

For the Women's, I'll speed over it.

The mean is 74.

The MAD is 3.

[tex]\underset{ \textit{angle in degrees} }{\textit{area of a regular polygon}}\\\\ A=\cfrac{ns^2}{4}\cot\left( \frac{180}{n} \right) ~~ \begin{cases} n=\stackrel{sides'}{number}\\ s=\stackrel{side's}{length}\\[-0.5em] \hrulefill\\ n=5\\ s=14 \end{cases}\implies A=\cfrac{(5)(14)^2}{4}\cot\left( \frac{180}{5} \right) \\\\\\ A=36\cot(20^o)\implies A\approx 98.91~m^2 \\\\\\ A=245\cot(36^o)\implies A\approx 337.21~in^2[/tex]

Make sure your calculator is in Degree mode.

The minimum value of the function on the interval [2, 11] is -100,000,000,000.

The function f(x) = -10ˣ is a decreasing exponential function, which means that its value decreases as x increases.

f(x) = -10ˣ on the interval [2, 11]

The minimum value of the function will be at the endpoint of the interval, which is at x=11.

f(11) = -10¹¹

f(11)= -100,000,000,000

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The average speed at which the canoe traveled toward the lighthouse is approximately 17.2 feet per minute.

A is the front of the canoe

B is the position of the observer in the lighthouse

AB is the distance between the front of the canoe and the observer

θ1 is the angle of depression at 5:00

θ2 is the angle of depression at 5:05

h is the height of the front of the canoe above the water (1.3 feet)

We want to find the speed at which the canoe is traveling toward the lighthouse, which we'll call x feet per minute.

Using trigonometry, we can find the following relationships:

tan(θ1) = h / AB          (1)

tan(θ2) = h / (AB + 5x)   (2)

We can rearrange equation (1) to solve for AB:

AB = h / tan(θ1)          (3)

Substituting equation (3) into equation (2), we get:

tan(θ2) = h / (h / tan(θ1) + 5x)

tan(θ2) = tan(θ1) / (1 + 5x * tan(θ1) / h)   (4)

We can solve equation (4) for x:

x = (tan(θ1) / (tan(θ2) - tan(θ1))) * h / 5   (5)

Now we just need to plug in the values we know:

θ1 = 6.5°

θ2 = 48.7° (since θ2 = θ1 + 42.2°)

h = 1.3 feet

Plugging these into equation (5), we get:

x = (tan(6.5°) / (tan(48.7°) - tan(6.5°))) * 1.3 / 5

x ≈ 17.2 ft/min

Therefore, the average speed at which the canoe traveled toward the lighthouse is approximately 17.2 feet per minute.

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The correct interpretation of the 95% confidence interval is that if Peggy were to repeat her sampling process many times and calculate a 95% confidence interval each time

Approximately 95% of those intervals would contain the true population means the height of young women aged 18 to 24 in the US. In other words, we can be 95% confident that the true population means height falls within the interval that Peggy calculated from her sample of 2,000 young women.

The correct interpretation is that there is a 95% chance that the true population mean height falls within the calculated interval.

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Answer:

the probability of picking two white socks is 1/3, or approximately 0.333.

Step-by-step explanation:

When choosing the first sock, there are 10 socks in total, and 6 of them are white. Therefore, the chance of picking a white sock on the first try is 6/10 or 3/5.

When picking the second sock, there are only 9 socks left in the drawer, since we did not replace the first sock. Of the remaining socks, 5 are white (since we did not replace the first white sock) and 4 are black. Therefore, the probability of picking a white sock on the second try, given that we picked a white sock on the first try, is 5/9.

Answer:  1/3

Step-by-step explanation:6*5/2 is number of ways that work. This is 15. There are 10*8/2=45 total ways. 15/45=1/3.

Answer:

Step-by-step explanation:

Sol'n,

Here,

The length of all sides of sq= height = h

Height of triangle=h

Base length of triangle=b

Now, We know that,

The entire figure is a trapezium,

so, Area of Trap.= 1/2 * h(length of diagonal one + length of diagonal 2)

or, 80 = 1/2 * h* {h +(b+h)}[ since here, the length of second diagonal is sum of the base and length of one side of sq]

or, 160 = h (2h+b)

2h^2 + hb - 160 = 0....(I)

Hence, I is the required eqn....

The Equation for Area of figure is area of Figure, h² + 1/2(h)(b) = 80

The measurement that expresses the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.

Given:

We have the figure consist of one square and one right triangle.

Now, Area of Figure,

= Area of square + area of Triangle

[tex]\dfrac{= \text{length} \times \text{width +}}{\times \text{base} \times \text{height}} =[/tex]

[tex]= 16 \times 12 + \dfrac{1}{2} \times 10 \times 20[/tex]

[tex]= 192+ 100[/tex]

[tex]=292 \ \text{unit}^2[/tex]

and, if the square with height, h, units and a right triangle with height, h units, and a base length, b units.

Then, area of Figure = h² + 1/2(h)(b) = 80 square units.

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