The half-life of radium is 1690 years. If 80 grams are present now, how much will be present in 430 years

The graph of the given inequality y ≥ x² + 1 is a shaded region above the downward-facing parabola -x² + 1.

Hence, graph A represents the given inequality.

The inequality  y ≥ x² + 1 represents a region in the coordinate plane where y is greater than or equal to the value of the function -x² + 1 for any given x. The graph of this inequality is a shaded region above the downward-facing parabola -x² + 1. The vertex of this parabola is located at the point (0,1), and as x moves away from 0, the value of the function becomes more negative.

Therefore, the shaded region includes all points (x, y) where y is greater than or equal to the y-value of the parabola at that x-value. The resulting graph is a curve that opens downward and flattens out at y=1 as x moves further away from 0.

Hence, the correct option is A.

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In order to find the probability of the intersection of two events, P(A ∩ B), we can use the formula: P(A ∩ B) = P(A) + P(B) - P(A ∪ B) Therefore, the correct answer is c. 0.2100.

In order to find the probability of the intersection of two events, P(A ∩ B), we can use the formula:

P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

Given the probabilities P(A) = 0.62, P(B) = 0.47, and P(A ∪ B) = 0.88, we can plug these values into the formula:

P(A ∩ B) = 0.62 + 0.47 - 0.88

P(A ∩ B) = 1.09 - 0.88

P(A ∩ B) = 0.21

Therefore, the correct answer is option (c) 0.2100.

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Answer:

Area = 2.1875

Step-by-step explanation:

The formula for area of a rhombus is A = 1/2(d1)(d2), where d1 is one of its rhombus and d2 is the other.  Thus, to find the area, we can plug into the formula 1.75 for d1 and 2.5 for d2 and solve for A:

A = 1/2(1.75)(2.5)

A = 0.875*2.5

A = 2.1875

a.) The hire purchase price would be =$3,933.6

b.) The amount of each monthly instalment would be =$218.5

C.) The difference between the price for hire purchase and cash price would be = $953.6

To hire purchase price for the video recorder would be = initial down payment + interest + cash price.

The initial down payment = 20/100×2980

= 59600/100 = $596

The interest = 15 % of outstanding balance

outstanding balance = 2980- 596 =$2,384

Therefore, 15/100 × 2,384

= 35760/100 = $357.60

Thet hire purchase price = 2980+596+357.60=$3,933.6

The amount for each month installment over the next 18 months = $3,933/18 = $218.5

The difference between hire purchase and cash price = $3,933.6- $2980 = $953.6

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The probability of randomly drawing a vowel is 2/5

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

W, E, L, O, V, E, M, A, T, H

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

Vowels = 4

Total = 10

So, we have

P(Vowel) = Vowel/Total

Substitute the known values in the above equation, so, we have the following representation

P(Vowel) = 4/10 = 2/5

Hence, the solution is 2/5

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The slices that will always result in a square plane section include;

The intersection of a three-dimensional object with a plane creates a two-dimensional shape known as a "plane section." When this section is square in shape, it is referred to as a "square plane section."

A cube possesses six faces that are all squares. Slicing the cube parallel to one of its sides produces a square plane section every time, given that the side being sliced also happens to be a square. Meanwhile, cutting the cube perpendicular to any of its other faces generates a comparable result: a square plane section.

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The dimensions of the new pyramid is 8 times the original volume.

The area of the pyramid's base and its height are exactly proportional to its volume, hence the volume of any pyramid is equal to the area of the base times the height of the pyramid divided by three.

Knowing the formula of the pyramid is:

1/3 x a x b x h

If the dimensions are doubled, it will be :

1/3 x 2a x 2b x 2h

So:

v2= 1/3 x 2a x 2b x 2h

v2 = 1/3 x 8 x a x b x h

Hence, The new volume is 8 times more than the original.

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The dimension of the rectangular prism is ( 2x - 1) which has a volume 4x⁴ + 14x³ - 8x²

The given volume of the rectangular prism is,

V = 4x⁴ + 14x³ - 8x²

Let the length, breadth and height of the rectangular prism be l, b, and h respectively.

Thus, by formula of volume of a rectangular prism we get,

l*b*h = 4x⁴ + 14x³ - 8x²

⇒ l*b*h = 2x² ( 2x² + 7x - 4)

= 2x² [ 2x² + 8x - x - 4]

= 2x² [ 2x( x + 4) -1( x + 4) ]

= 2x² ( 2x - 1 )( x + 4 )

Therefore, by equating the above equation, obtained from simplifying the equation of volume of a rectangular prism, with zero , we get,

2x² ( 2x - 1 )( x + 4 ) = 0

⇒ 2x² = 0 ⇒ x = 0

and, ⇒ 2x - 1 = 0 ⇒ x = 1/2

and, ⇒ x + 4 = 0 ⇒ x = -4

Thus we can see that only equation (2x - 1) gives a possible value of x, that is either the length or breadth or height of the rectangular prism.

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The Fourier series for the original function f(x) is:

f(x) = (1/2) + ∑[n=1,∞] [(4/nπ²) * (1 - (-1)^n) cos(nπx/2) + (4/nπ) * sin(nπ/2) sin(nπx/2)] for x € (-1,1)

To find the Fourier series coefficients for the given function, we need to first determine the period of the function.

Since the function is defined differently for x in the interval (-1,0) and (0,1), we can break down the function into two separate periodic functions, each with its own period.

For the interval (-1,0), the function is a constant function equal to 1. Hence, the period is simply 2.

For the interval (0,1), the function is a linear function given by f(x) = 2 - x. The period of a linear function is always infinite, but we can restrict the domain to a smaller interval to get a periodic function. We can choose the interval (0,2) as the period for this function, since f(x + 2) = 2 - (x + 2) = 2 - x = f(x) for all x in the interval (0,1).

Now, we can write the Fourier series for each of the two periodic functions:

For the function defined on (-1,0), the Fourier series coefficients are given by:

an = (1/2) * ∫[-1,1] f(x) cos(nπx/2) dx

= (1/2) * ∫[-1,0] cos(nπx/2) dx (since f(x) = 1 for x in (-1,0))

= (1/2) * [(2/π) * sin(nπ/2)]

bn = (1/2) * ∫[-1,1] f(x) sin(nπx/2) dx

= (1/2) * ∫[-1,0] sin(nπx/2) dx (since f(x) = 1 for x in (-1,0))

= (1/2) * [(2/π) * (1 - cos(nπ/2))]

For the function defined on (0,1), the Fourier series coefficients are given by:

an = (1/2) * ∫[0,2] f(x) cos(nπx/2) dx

= (1/2) * ∫[0,1] (2 - x) cos(nπx/2) dx

= (1/2) * [(4/nπ²) * (1 - (-1)^n)]

bn = (1/2) * ∫[0,2] f(x) sin(nπx/2) dx

= (1/2) * ∫[0,1] (2 - x) sin(nπx/2) dx

= (1/2) * [(4/nπ) * sin(nπ/2)]

Hence, the Fourier series for the original function f(x) is:

f(x) = (1/2) + ∑[n=1,∞] [(4/nπ²) * (1 - (-1)^n) cos(nπx/2) + (4/nπ) * sin(nπ/2) sin(nπx/2)] for x € (-1,1)

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Where the logistic growth function is f( t) = 116000/(1+5200e⁻t)

a) 22 people were initially affected

b) after 4 weeks the numbers increased to 1,205 approximately

c) the limiting size of the population that becomes ill is 116,000.

The logistic equation (also known as the Verhulst model or logistic growth curve) is a population growth model developed by Pierre Verhulst (1845, 1847).

a) since when the epidemic began, the initial number of infection will always be zero, thus:

f( t) = 116000/(1+5200e⁻t )

Note that e is the mathematical constant known as Euler's number or the natural base. e = 2.71828

f(0) = 116000/(1+5200e⁻0 )

f(0) = 116000/(1+5200 (2.71828)⁻0 )

f(0) = 116000/(1+5200 (1) )

f(0) = 116000/(1+5200 )

f(0) = 116000/(5201 )

f(0) = 22.3034031917

f (0) [tex]\approx[/tex] 22

B) by the fourth week,

the expression became f(4) = 116000/(1+5200e⁻4 )

f (4) = 116000 /(1+5200 (e)⁻4 )
f (4) = 1205.30347384
f (4)  [tex]\approx[/tex] 1205

C)

Since the logistic curve is given as....

f(t) = 116000/(1+5200e⁻t )

as t becomes smaller and smaller nearing 0, the denominator will be almost 1

So
f(t) = 116000/(1+5200e⁻ ∞ )

f(t) = 116000/(1+0 )

f(t) = 116,000

As you can see, the limit to the size of the population that can fall ill is 116,000 peple.

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The change in the advertising budget necessary to maintain a monthly profit of $93,500 if the insurance company hires 5 new sales associates is estimated to be an increase of $2,308.

To estimate the change in advertising budget necessary to maintain a monthly profit of $93,500 with the addition of 5 new sales associates, we need to consider the potential impact of these new hires on the company's revenue and expenses.

Assuming that each new sales associate is expected to generate $10,000 in revenue per month, the total increase in revenue would be $50,000 (5 sales associates x $10,000).

However, there may also be additional expenses associated with the new hires, such as salaries and benefits. Let's assume that the total cost of adding 5 new sales associates is $30,000 per month.

Therefore, the net increase in monthly profit due to the new hires would be $20,000 ($50,000 in revenue - $30,000 in expenses).

To maintain a monthly profit of $93,500, the company would need to increase its advertising budget by an amount that would generate an additional $20,000 in profit. Assuming that the company's profit margin on its products is 20%, this would require an additional $100,000 in revenue per month ($20,000 / 0.20).

Based on the historical data and market conditions, the company could estimate the additional advertising budget required to generate this amount of revenue.

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The standard error of the mean is 1.16. At 95% confidence, the margin of error is 2.27.

(a) The standard error of the mean, σx, can be calculated using the formula:
σx = σ/√n
where σ is the population standard deviation, and n is the sample size.
Substituting the given values, we get:
σx = [tex]\frac{9}{ \sqrt{60} }[/tex]
σx = 1.16
Therefore, the standard error of the mean is 1.16.

(b) To find the margin of error, we need to use the formula:
The margin of error = z(σx)
where z is the z-score corresponding to the level of confidence. For 95% confidence, the z-score is 1.96 (using a standard normal distribution table).
Substituting the values we get:
Margin of error = 1.96(1.16)
Margin of error = 2.27

Therefore, at 95% confidence, the margin of error is 2.27. This means that we can be 95% confident that the true population means falls within 2.27 units of the sample mean of 25.

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Answer:3 1/2 times 5= 15  1/10 - 7 1/4 = 8 6/40 or 8 3/20

Step-by-step explanation:

Answer:

3/4 or 0.75 tons

Step-by-step explanation:

The sum of the two months is 7 1/4 which is 29/4. For the first month, the company used 3 1/2 (7/2 or 14/4)

(29-14)/4=15/4.

Thus the second month you use 15/4.

Don’t simplify it yet-
15/4 divided by 5 is 3/4.

The answer is option(b) greater than equal to the approximation given by Chebyshev's theorem.

To answer this, we need to use the empirical rule and compare it with the approximation given by Chebyshev's theorem.

The empirical rule states that approximately 68% of the data will be within one standard deviation, 95% within two standard deviations, and 99.74% within three standard deviations of the mean, given that the data has a bell-shaped distribution.

In this case, we're looking at 3 standard deviations from the mean. According to the empirical rule, approximately 99.74% of the data will be within 3 standard deviations. Now, we compare the approximation given by the empirical rule (99.74%) to the approximation given by Chebyshev's theorem (89%).

Since 99.74% (empirical rule) is greater than 89% (Chebyshev's theorem), the approximation given by the empirical rule is greater than equal to the approximation given by Chebyshev's theorem.

Your answer: B. greater than equal to the approximation given by Chebyshev's theorem.

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The compression technique encodes the digital value of an analog sample, based on the change from the previous sample is c. Differential PCM using delta encoding

By accounting for the difference or change between successive samples, Differential Pulse Code Modulation (DPCM) is a compression method used to encode digital values of analogue samples. In DPCM, the delta, or difference between the current and previous samples, is quantized and encoded, which results in a less amount of data than if the samples' absolute values were simply recorded.

It is a method that takes use of the correlation or resemblance between successive samples in a variety of analogue signals. But it might experience error propagation, where mistakes in the decoded delta values can build up over time and result in a deterioration in signal quality.

Complete Question:

Which compression technique encodes the digital value of an analog sample, based on the change from the previous sample?

a. LZ78 compression

b. Shannon-Fano encoding

c. Differential PCM using delta encoding

d. Huffman coding

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Answer:
The equation y = 75 - 2.85x represents the amount of money left on the gift card, where x is the number of Chai lattes Angie has purchased and y is the remaining gift card balance.

To find the x-intercept, we set y = 0 and solve for x:

0 = 75 - 2.85x

2.85x = 75

x = 26.32 (rounded to the nearest tenth)

So the x-intercept is approximately 26.3 Chai lattes, meaning that Angie can buy a maximum of 26 Chai lattes with her gift card.

To find the y-intercept, we set x = 0 and solve for y:

y = 75 - 2.85(0)

y = 75

So the y-intercept is $75, meaning that when Angie has not yet bought any Chai lattes, her gift card balance is $75.

To predict how many Chai lattes Angie has purchased when her gift card balance is about $35, we set y = 35 and solve for x:

35 = 75 - 2.85x

2.85x = 40

x = 14.04 (rounded to the nearest tenth)

So when Angie's gift card balance is about $35, she has purchased approximately 14 Chai lattes.

The ratio StartFraction 14 Over 35 EndFraction is equivalent to the other three ratios, and the ratio that is different from the others is StartFraction 8 Over 20 EndFraction.

To determine which ratio is not equivalent to the others, we need to simplify each ratio to its lowest terms.

Give the following ratios :

[tex]2 / 5 = 6 /10 = 8 / 20 = 12 / 30.[/tex]

- StartFraction 2 Over 5 EndFraction: This ratio is already in its simplest form.

- StartFraction 6 Over 10 EndFraction: We can simplify this ratio by dividing both the numerator and denominator by their greatest common factor (GCF), which is 2.

-StartFraction 6 Over 10 EndFraction = StartFraction 3 Over 5 EndFraction

- StartFraction 8 Over 20 EndFraction: We can simplify this ratio by dividing both the numerator and denominator by their GCF, which is 4.

StartFraction 8 Over 20 EndFraction = StartFraction 2 Over 5 EndFraction

- StartFraction 12 Over 30 EndFraction: We can simplify this ratio by dividing both the numerator and denominator by their GCF, which is 6.

StartFraction 12 Over 30 EndFraction = StartFraction 2 Over 5 EndFraction

Therefore, the ratios StartFraction 6 Over 10 EndFraction, StartFraction 8 Over 20 EndFraction, and StartFraction 12 Over 30 EndFraction are all equivalent to StartFraction 2 Over 5 EndFraction. The ratio that is different from the others is StartFraction 14 Over 35 EndFraction, which can be simplified by dividing both the numerator and denominator by their GCF, which is 7.

[tex]StartFraction 14 Over 35 EndFraction = StartFraction 2 Over 5 EndFraction.[/tex]

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The appropriate alternative hypothesis is: Hₐ: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] does not equal 0, where [tex]\mu_{sc[/tex] is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme, and [tex]\mu_{dc[/tex] is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

An assertion used in statistical inference experiments is known as the alternative hypothesis. It is indicated by Hₐ or H₁ and runs counter to the null hypothesis.

The appropriate alternative hypothesis is: Hₐ: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] does not equal 0, where [tex]\mu_{sc[/tex] is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme, and [tex]\mu_{dc[/tex] is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

This hypothesis is appropriate because it is testing whether there is a statistically significant difference in the mean time to complete all matches between the two matching schemes. The null hypothesis would be that there is no difference in the mean time between the two schemes, i.e., H₀: [tex]\mu_{sc[/tex] - [tex]\mu_{dc[/tex] = 0.

To test this hypothesis, we can use a paired t-test, which compares the mean difference between the two sets of measurements (in this case, the time to complete all matches using the two matching schemes) to the standard error of the mean difference. If the t-test results in a p-value that is smaller than the chosen significance level (typically 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between the mean times for the two matching schemes.

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The appropriate alternative hypothesis is: Hₐ:  - does not equal 0, where  is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme and  is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

An assertion used in statistical inference experiments is known as the alternative hypothesis. It is indicated by Hₐ or H₁ and runs counter to the null hypothesis.

The appropriate alternative hypothesis is: Hₐ:  - does not equal 0, where  is the mean time to complete all matches using the "shapes and cutouts are all the same color" matching scheme and is the mean time to complete all matches using the "shapes and cutouts are different colors" matching scheme.

This hypothesis is appropriate because it is testing whether there is a statistically significant difference in the mean time to complete all matches between the two matching schemes. The null hypothesis would be that there is no difference in the mean time between the two schemes, i.e., H₀:  - = 0.

To test this hypothesis, we can use a paired t-test, which compares the mean difference between the two sets of measurements (in this case, the time to complete all matches using the two matching schemes) to the standard error of the mean difference. If the t-test results in a p-value that is smaller than the chosen significance level (typically 0.05), we reject the null hypothesis and conclude that there is a statistically significant difference between the mean times for the two matching schemes.

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Answer:

2nd and 4th

Step-by-step explanation:

In the second quadrant, the x-coordinate is negative and the y-coordinate is positive. If we let the x-coordinate be -a, then the y-coordinate will be a. Therefore, the point will have the form (-a, a), and the y-coordinate will be the opposite of the x-coordinate.

In the fourth quadrant, the x-coordinate is positive and the y-coordinate is negative. If we let the x-coordinate be a, then the y-coordinate will be -a. Therefore, the point will have the form (a, -a), and again, the y-coordinate will be the opposite of the x-coordinate.

The explicit rule for sequence 3200,1600,800, 400,... is aₙ = 3200(1/2)⁽ⁿ⁻¹⁾, the explicit rule for sequence 12, 84, 588, 4116, .. is aₙ = 12(7)⁽ⁿ⁻¹⁾, the explicit rule for sequence 1395, 465, 155, 51.67,... is aₙ = 1395(1/3)⁽ⁿ⁻¹⁾.

The common ratio in this geometric sequence is 1/2. Thus, the explicit rule for this sequence is given by

aₙ = 3200(1/2)⁽ⁿ⁻¹⁾

where aₙ represents the nth term of the sequence.

This sequence appears to be a geometric sequence where the common ratio is 7. Thus, the explicit rule for this sequence is

aₙ = 12(7)⁽ⁿ⁻¹⁾

where aₙ represents the nth term of the sequence.

This sequence appears to be a geometric sequence where the common ratio is 1/3. Thus, the explicit rule for this sequence is

aₙ = 1395(1/3)⁽ⁿ⁻¹⁾.

where aₙ represents the nth term of the sequence.

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Answer:

65%

Step-by-step explanation:

ITS CORRECT

Answer:[tex]\left(x-14\right)^{2}+\left(y-15\right)^{2}=92[/tex]

Step-by-step explanation:

Since the x value is 14 and the y value is 15, we know our H and K for this formula. In the formula, we state that the first part of the formula is equal to the radius squared. That would look something like this without the filled in values:

[tex]\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}[/tex]

The center point of the circle is found at (h, k)

I hope this helps :)

To design the optimal contract that induces both workers to exert effort all the time, consider the following steps:

1. Determine the joint probabilities of success for each combination of efforts:
  - Both workers shirk: Probability of success is p0.
  - One worker shirks and the other works hard: Probability of success is p1.
  - Both workers work hard: Probability of success is p2.

2. Identify the complementarity or substitutability of workers' efforts:
  - Since p2 > p1 > p0, the workers' efforts are complementary. This means that the success probability increases when both workers exert effort, as compared to only one worker doing so.

3. Design the optimal contract based on complementarity:
  - The principal should offer a contract with a bonus B, paid only if the project is successful.
  - To incentivize both workers to exert effort, the bonus should satisfy the following condition:
    B > 2c / (p2 - p1)

This ensures that the benefit of exerting effort (i.e., receiving the bonus) outweighs the cost of effort (c) for both workers. Since the workers' efforts complement each other, they will be more likely to exert effort knowing that their combined efforts increase the probability of project success and receiving the bonus.

In summary, the optimal contract should offer a bonus B that satisfies B > 2c / (p2 - p1) and is paid only upon project success. This contract incentivizes both workers to exert effort all the time, as their efforts complement each other and increase the probability of project success.

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Answer:

2(9(5) + 5(3) + 9(3)) = 2(45 + 15 + 27) = 2(87) = 174 square inches

Four 4cm pipes would give the same amount of water as one 8cm pipe.

We have,

The volume of water that can flow through a pipe is directly proportional to the cross-sectional area of the pipe, which is proportional to the square of its radius.

The area of the 8cm pipe is A1 = π(8cm)²

= 64π cm².

Let x be the number of 4cm pipes that would be needed to replace the 8cm pipe.

The area of one 4cm pipe is A2 = π(4cm)² = 16π cm².

Therefore, the area of x 4cm pipes is Ax = x(16π) = 16xπ cm².

We want to find the value of x such that Ax = A1.

That is, 16xπ = 64π

Solving for x, we get:

x = (64π) / (16π) = 4

Therefore,

Four 4cm pipes would give the same amount of water as one 8cm pipe.

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The area of the gazebo is approximately 127.23 square feet.

To find the actual diameter of the gazebo, we need to use the scale of 2 inches = 5 feet. This means that for every 2 inches on the blueprint, the actual distance is 5 feet. Therefore, we can set up a proportion:

2 inches / 5 feet = 5.1 inches / x

where x is the actual diameter of the gazebo.

Solving for x, we get:

x = (5.1 inches * 5 feet) / 2 inches

x = 12.75 feet

So the actual diameter of the gazebo is 12.75 feet.

To find the area of the gazebo, we need to use the formula for the area of a circle:

A = π[tex]r^2[/tex]

where r is the radius of the circle. Since we know the diameter is 12.75 feet, the radius is half of that:

r = 12.75 feet / 2

r = 6.375 feet

Now we can plug in the radius to the formula for the area:

A = π(6.375 feet[tex])^2[/tex]

A = 127.23 square feet

So the area of the gazebo is approximately 127.23 square feet.

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The inverse of f(x) is: [tex]f^{-1(x)}[/tex] = (x - 8)/5 switched the variables, so instead of [tex]f^{-1(x)}[/tex] , we wrote it as [tex]f^{-1(x)}[/tex] .

What is function?

In mathematics, a function is a relationship between two sets of elements, called the domain and the range, such that each element in the domain is associated with a unique element in the range.

To find the inverse of the function f(x) = 8 + 5x, we need to solve for x in terms of f(x).

Let y = f(x) = 8 + 5x

To find the inverse, we need to isolate x on one side of the equation:

y = 8 + 5x

y - 8 = 5x

x = (y - 8)/5

Therefore, the inverse of f(x) is:

[tex]f^{-1(x)}[/tex] = (x - 8)/5

Or, we can write it as:

[tex]f(x)^{-1}[/tex] = (x - 8)/5

Note that we switched the variables, so instead of [tex]f^{-1}[/tex] , we wrote it as [tex]f^{-1(x)}[/tex] .

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The percent chance that the ball you draw is white is 70%.

To find the percent chance of drawing a white ball, we will calculate the probability and then convert it into a percentage.

1. First, determine the total number of balls in the bag: 7 white balls + 3 red balls = 10 balls.
2. Next, determine the number of favorable outcomes, which is the number of white balls: 7 white balls.
3. Calculate the probability by dividing the number of favorable outcomes by the total number of balls: 7 white balls ÷ 10 balls = 0.7.
4. Convert the probability to a percentage by multiplying it by 100: 0.7 x 100 = 70%.

So, the percent chance that the ball you draw is white is 70%.

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