A tank in the form of a right-circular cylinder standing on end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank in feet after t seconds is described by dh dt А. 2gh AW where A and A, are the cross-sectional areas of the water and the hole in square feet, respectively. (a) Solve for h(t) if the initial height of the water is H. Give its interval I of definition in terms of the symbols Awr An, and H. Use g = 32 ft/s2. (4„VH – 44,4) h(t) = osts 4A 4VĀ By hand, sketch the graph of h(t). h h H H/2 t h h H/2H H t (b) Suppose the tank is 11 ft high and has radius 2 ft and the circular hole has radius in. If the tank is initially full, how long will it take to empty? (Round your answer to two decimal places.) sec

The answer of the question based on expressing the side of the base of a rectangular prism in terms of V and b the answer is the side of the base a is equal to the square root of the volume V divided by the height b.

Volume is measure of  amount of space that  three-dimensional object takes up. It is typically expressed in units of cubic length, like cubic meters, cubic feet, or cubic centimeters.

The formula for volume of many common geometric shapes can be calculated using mathematical formulas, while volumes of irregular shapes can be determined by measuring  amount of water they displace or by using advanced imaging techniques like CT scans.

The formula for  volume of  rectangular prism are:

V = length×width×height

Since the base of the prism is a square, the length and width are both equal to a. Therefore:

V = a × a × b

Simplifying this expression:

V = a²b

Dividing both sides by b:

a² = V/b

Taking the square root of both sides:

a = √(V/b)

Therefore, the side of the base a is equal to the square root of the volume V divided by the height b.

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The probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election is 0.0532.

To calculate the probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election, we can use the binomial probability formula.

The formula is: P(x) = (ⁿCₓ) × pˣ × (1-p)⁽ⁿ⁻ˣ⁾

In this case, n = 328, p = 46.37%, and x = 164 (since half of 328 is 164).

Plugging in the numbers we get:

P(x) = ³²⁸C₁₆₄ × (0.4637)¹⁶⁴ × (0.5363)⁽³²⁸⁻¹⁶⁴⁾ = 0.0532

Therefore, the probability that at least half of the 328 voters contacted in the telephone survey supported the current governor in the last election is 0.0532.

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Answer:

[tex]\frac{-1}{3}[/tex]

Step-by-step explanation:

Look at the red slope triangle that is drawn. If you start at the point on the left, you go down 1 unit and then to the right 3 units.  This would be represented by -1/3.  Down would be negative and going right would be positive.  The slope is the rise over the run.

Helping in the name of Jesus.

Answer:

y = -2x + 12

Step-by-step explanation:

In order to put the equation in slope intercept form [tex]y=mx+b[/tex] we need to solve for y.

[tex]4x+2y=24[/tex]

Subtract 4x on both sides

[tex]2y=-4x+24[/tex]

Divide by 2

[tex]y=-2x+12[/tex]

Answer:

If f(x)=x^3, evaluate f(x+h)-f(x)÷h, Where h* 0. Use your result to find the derivative of f(x) with respect to x. Differentiate with respect to x (x²-3x+5)(2x-7) .Find with respect to x the derivative of sinx ÷1– cosx​

Step-by-step explanation:

We are given f(x) = x^3. We need to find the value of (f(x+h) - f(x))/h.

f(x+h) = (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3

Therefore, (f(x+h) - f(x))/h = [x^3 + 3x^2h + 3xh^2 + h^3 - x^3]/h

= 3x^2 + 3xh + h^2

Taking the limit of the above expression as h approaches 0, we get:

lim(h→0) [(f(x+h) - f(x))/h] = 3x^2

Therefore, the derivative of f(x) = x^3 with respect to x is 3x^2.

Next, we need to differentiate (x^2-3x+5)(2x-7) with respect to x.

Using the product rule, we get:

d/dx [(x^2-3x+5)(2x-7)] = (2x-7)(2x-3) + (x^2-3x+5)(2)

Simplifying, we get:

d/dx [(x^2-3x+5)(2x-7)] = 4x^2 - 20x + 11

Therefore, the derivative of (x^2-3x+5)(2x-7) with respect to x is 4x^2 - 20x + 11.

Finally, we need to find the derivative of sin(x)/(1-cos(x)) with respect to x.

Using the quotient rule, we get:

d/dx [sin(x)/(1-cos(x))] = [(1-cos(x))cos(x) - sin(x)(sin(x))]/(1-cos(x))^2

Simplifying, we get:

d/dx [sin(x)/(1-cos(x))] = cosec(x/2)^2

Therefore, the derivative of sin(x)/(1-cos(x)) with respect to x is cosec(x/2)^2.

(a) The null hypothesis is that p = 0.5, and the alternative hypothesis is that p ≠ 0.5.

To conduct a hypothesis test, we need to define the null and alternative hypotheses. Let p be the proportion of US adults who believe raising the minimum wage will help the economy. The null hypothesis is that p = 0.5, and the alternative hypothesis is that p ≠ 0.5.

(b) The conditions for using a normal approximation to the binomial distribution are met since np = 1000(0.5) = 500 and n(1-p) = 1000(0.5) = 500 are both greater than 10.

(c) Using a significance level of α = 0.05, the test statistic is z = (0.42 - 0.5) / sqrt[(0.5)(0.5)/1000] = -2.38. The corresponding p-value is P(z < -2.38) = 0.017.

(d) Since the p-value is less than the significance level, we reject the null hypothesis. There is evidence to suggest that the proportion of US adults who believe raising the minimum wage will help the economy is not equal to 0.5. Therefore, we can conclude that a majority of US adults do not believe raising the minimum wage will help the economy.

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The linear approximation of the given function f(x) is 40.5.

As f(X) = 4√(81+x) so by L(x) = f(a) + f'(a)(x-a) where a=0 and f(x)=4√(81+x) Then f'(x) = (1/2)(81+x)^(-1/2) So, f'(a) = (1/2)(81)^(-1/2) = 1/18.

Thus, L(x) = f(0) + f'(0)(x-0)

L(x) = 4√(81) + (1/18)x

L(x) = 36 + (1/18)x

Now we are asked to approximate the values 4√81.02 and 4√80.99 by using the linear approximation we found above. Let's solve the problems. Approximation of 4√81.02.L(x) = 36 + (1/18)x Now x=81.02. Thus, L(81.02) = 36 + (1/18)81.02L(81.02) = 36 + 4.5L(81.02) = 40.5.

Thus, the linear approximation of 4√81.02 is 40.5. Approximation of 4√80.99.L(x) = 36 + (1/18)x Now x=80.99. Thus, L(80.99) = 36 + (1/18)80.99L(80.99) = 36 + 4.49444...L(80.99) = 40.49444... ≈ 40.5. Thus, the linear approximation of 4√80.99 is approximately 40.5.

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The answer is: A) r=0.04; R=0.96 B) 8% C) 92.16%.

The Hardy-Weinberg equilibrium is a method for determining the frequency of certain genetic traits in a population. The following are the frequencies of the red hair (r) and non-red hair (R) alleles in the Norwegian population, according to the question: A) The total frequency of alleles is 1. If red hair (r) is found in approximately 4% of the people in Norway, then the frequency of the R allele must be 0.96 or 96%. R = 0.96 r = 0.04 B) The frequency of hetero zygotes in a population can be determined by multiplying the frequency of the R allele by the frequency of the r allele and then multiplying that number by 2.

Heterogeneous genotype = 2pq Here, p represents the frequency of the R allele, and q represents the frequency of the r allele. p = R = 0.96 q = r = 0.04 Therefore, 2pq = 2(0.96 x 0.04) = 0.077, or approximately 8%. C) In order for a child to have red hair, both parents must carry the r allele. If both parents are homozygous for the R allele, there is no chance that their child will have red hair. This means that the proportion of matings that cannot result in a child with red hair is (0.96 x 0.96) = 0.9216 or 92.16%.

Therefore, the answer is: A) r=0.04; R=0.96 B) 8% C) 92.16%.

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So, the correct option is (B) the function. [tex](f - g)(x)[/tex] is equal to the square root of[tex]x/8[/tex] minus [tex]2/x[/tex] plus 6.

A function is a rule or mapping that associates each input value from a set (called the domain) with a unique output value from another set (called the range or codomain).

For example, let's consider a function.[tex]f(x) = x^2[/tex]. Here, the domain of the function could be any set of real numbers, and the range would be the set of non-negative real numbers. For any given input value x, the function f(x) would return the output value. [tex]x^2.[/tex]

by the question.

The function (f - g) (x) is defined as the difference between f(x) and g(x) evaluated at x. Therefore:

[tex](f - g)(x) = f(x) - g(x)[/tex]

Substituting the given expressions for f(x) and g(x) into this formula, we get:

[tex](f - g)(x) = [\sqrt(x/8) + 11] - [5 + 2/x][/tex]

Simplifying this expression, we can combine like terms to get:

[tex](f - g)(x) = \sqrt(x/8) - 2/x + 6[/tex]

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This type of growth model is often used to model revenue or sales of a product, as these types of processes often grow in a linear fashion.

Growth models, like those used in ForecastX, are typically used to model situations where a process grows at an exponential rate. This is a type of growth where the rate of increase is proportional to the size of the process, meaning that the process increases quickly over time.

This type of growth model is often used to model population growth, technological growth, and financial growth. In these cases, the growth model is usually used to make predictions about future growth in the process.  

In some cases, a growth model can be used to model a process that grows in a more or less constant rate. In this situation, the rate of increase is not proportional to the size of the process, and the rate of increase remains relatively stable over time. This type of growth model is often used to model natural resources, as these resources typically increase or decrease at a steady rate.

Finally, growth models can also be used to model a process that grows in a linear fashion. In this case, the rate of increase is constant and increases in a straight line over time.

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Answer:

(2x - y)(3b + c + 2d)

Step-by-step explanation:

Factorize:

From the first two terms 6bx and (-3by) take out the common factor 3b.

From the third and fourth term 2cx and (-cy) take out the common factor c.

From the fifth and sixth term 4dx and (-2dy) take out the common factor 2d.

6bx - 3by  + 2cx - cy + 4dx - 2dy = 3b(2x - y) + c(2x - y) + 2d(2x - y)

                                                      = (2x - y)(3b + c + 2d)

Answer: 12.84

Step-by-step explanation:

if x = 3.2 and y = 6.1  and Z = 0.2

then plug in the numbers

(3.2)(0.2) + (6.1)(2)

0.64 + 12.2 = 12.84

Any variable next to a number means multiplication.

if I was wrong lmk

The missing amounts for Walco Corporation and Gunther Enterprises assuming  all changes in stockholders equity are due to changes in retained earnings are
(a) Walco Corporation Total Stockholders' Equity =  $54,000
(b) Walco Corporation Total Assets = $182,000
(c) Walco Corporation Dividends = $47,100
(d) Gunther Enterprises Total Liabilities = $140,000
(e) Gunther Enterprises Total Stockholders' Equity = $91,500
(f) Gunther Enterprises Total Revenues =  $135,100

A balance sheet is a financial statement that reports a company's assets, liabilities, and stockholder equity on a specific date. Assets are resources a company owns that have monetary value, liabilities are obligations that must be paid in the future, and stockholder equity is the difference between a company's assets and liabilities. To calculate the missing amounts, you need to subtract the beginning of year figures from the end of year figures.

a) Total liabilities + Total stockholders' equity = Total assets

Total liabilities + $54,000 = $100,000

Total liabilities = $46,000

(b) Total assets = Total liabilities + Total stockholders' equity

Total assets = $128,000 + $54,000

Total assets = $182,000

(c) Changes during year in retained earnings = Total revenues - Total expenses - Dividends

Changes during year in retained earnings = $219,000 - $167,000 - $4,900

Changes during year in retained earnings = $47,100

The missing values for Gunther Enterprises:

(d) Total liabilities + Total stockholders' equity = Total assets

$67,500 + $(e) = $159,000

$(e) = $91,500

(f) Changes during year in retained earnings = Total revenues - Total expenses - Dividends

Changes during year in retained earnings = $(f) - $79,000 - $4,900

Changes during year in retained earnings = $(f) - $83,900

Using the balance sheet equation, we can find the missing values:

(d) Total liabilities = Total assets - Total stockholders' equity

Total liabilities = $159,000 - $67,500

Total liabilities = $91,500

(e) Total stockholders' equity = Total assets - Total liabilities

Total stockholders' equity = $190,000 - $50,000

Total stockholders' equity = $140,000

(f) Changes during year in retained earnings = Total revenues - Total expenses - Dividends

Changes during year in retained earnings = $219,000 - $79,000 - $4,900

Changes during year in retained earnings = $135,100

Therefore, the missing amounts are:

(a) Walco Corporation Total Stockholders' Equity =  $54,000
(b) Walco Corporation Total Assets = $182,000
(c) Walco Corporation Dividends = $47,100
(d) Gunther Enterprises Total Liabilities = $140,000
(e) Gunther Enterprises Total Stockholders' Equity = $91,500
(f) Gunther Enterprises Total Revenues =  $135,100

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A system of linear inequalities that the graph represent include the following:

x ≥ 4.

y < -x - 2

y ≥ 3x + 3

y > x - 4

In Mathematics, the slope-intercept form of the equation of a straight line is given by this mathematical expression;

y = mx + c

Where:

At data points (-3, 0) and (0, 3), the slope of this line can be calculated as follows;

Slope = (3 - 0)/(0 + 3) = 3/3 = 1

y-intercept = 3.

Therefore, a linear inequality that models the line is given by:

y ≥ 3x + 3

At data points (-2, 0) and (0, -2), the slope of this line can be calculated as follows;

Slope = (-2 - 0)/(0 + 2) = -2/2 = -1

y-intercept = -2.

Therefore, a linear inequality that models the line is given by:

y < -x - 2

At data points (4, 0) and (0, -4), the slope of this line can be calculated as follows;

Slope = (-4 - 0)/(0 - 4) = -4/-4 = 1

y-intercept = -4.

Therefore, a linear inequality that models the line is given by:

y > x - 4

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Yes, the triangle is a right triangle, the correct option is A.

We can see that the vertices of the triangle are at:

A = (6, 4)

B = (-6, 9)

C = (0, 0)

Then the lenghts of each of the sides is:

AC = √( (6 - 0)² + (4 - 0)²) = √52

BC = √( (-6 - 0)² + (9 - 0)²) = √117

AB = √( (6 + 6)² + (4 - 9)²) = 13

If this is a right triangle, then the pythagorean theorem must be true, so:

AB² = AC² + BC²

13² =  √52²  + √117²

169 = 52 + 117

169 = 169

This means that BC is perpendicular to AC.

So yes, it is a right triangle. The correct option is A.

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Answer:

Step-by-step explanation:

Question: The heights of adult men can be approximated as normal, with a mean of 70 and a standard deviation of 3, the probability that a man is shorter than 65 inches is approximately 0.0475 or 4.75%.

Let X be the height of an adult man, which follows a normal distribution with mean μ = 70 and standard deviation σ = 3. Then, we need to find the probability that a man is shorter than some height, say x₀. We can write this probability as P(X < x₀).To find P(X < x₀), we need to standardize the random variable X by subtracting the mean and dividing by the standard deviation. This yields a new random variable Z with a standard normal distribution. Mathematically, we can write this transformation as:Z = (X - μ) / σwhere Z is the standard normal variable.

Now, we can find P(X < x₀) as:P(X < x₀) = P((X - μ) / σ < (x₀ - μ) / σ) = P(Z < (x₀ - μ) / σ)Here, we use the fact that the probability of a standard normal variable being less than some value z is denoted as P(Z < z), which is available in standard normal tables.

Therefore, to find the probability that a man is shorter than some height x₀, we need to standardize the height x₀ using the mean μ = 70 and the standard deviation σ = 3, and then look up the corresponding probability from the standard normal table.In other words, the probability that a man is shorter than x₀ can be expressed as:P(X < x₀) = P(Z < (x₀ - 70) / 3)We can now use standard normal tables or software to find the probability P(Z < z) for any value z.

For example, if x₀ = 65 (i.e., we want to find the probability that a man is shorter than 65 inches), then we have:z = (65 - 70) / 3 = -1.67Using a standard normal table, we can find that P(Z < -1.67) = 0.0475. Therefore, the probability that a man is shorter than 65 inches is approximately 0.0475 or 4.75%. Thus, P(X < 65) = 0.0475 or 4.75%. Therefore, the probability that a man is shorter than 65 inches is approximately 0.0475 or 4.75%.

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You have given a relation R = {(x,y): y is the area of triangle x 3}. This means that for each value of x, which represents the base of a triangle, y is the area of that triangle with height 3.

To find out if this relation is a function, we need to check if there are any repeated x-values with different y-values. If there are none, then it is a function.

One way to do this is to use the formula for the area of a triangle: A = (1/2)bh, where b is the base and h is the height. Using this formula, we can find the y-values for some x-values:

[tex]x | y| - 0 | 0 1 | (1/2) * 1 * 3 = 3/2 2 | (1/2) * 2 * 3 = 3 3 | (1/2) * 3 * 3 = 9/2 4 | (1/2) * 4 * 3 = 6[/tex]

Therefore, As you can see, there are no repeated x-values with different y-values, this relation R is a function.

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Therefore , the solution of the given problem of unitary method comes out to be it weighs 0.01 times as much as the first nugget.

This common convenience, already-existing variables, or all important elements from the original Diocesan customizable survey that followed a particular event methodology can all be used to achieve the goal. If it does, there will be another chance to get in touch with the entity. If it doesn't, each of the crucial elements of a term proof outcome will surely be lost.

Here,

We can divide the weight of the second nugget by the weight of the first nugget to determine how many times as much the second nugget weights the first:

=> 0.8 oz / 0.008 oz = 100

The second piece is therefore 100 times heavier than the first.

We can divide the first nugget's weight by the second nugget's weight to determine how much the first nugget weights in relation to the second nugget:

=> 0.008 oz /0.8 oz = 0.01

In other terms, the second nugget weighs 100 times as much as the first nugget, or it weighs 0.01 times as much as the first nugget.

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Answer:

16x-20

Step-by-step explanation:

6(2x-3)-2(2x+1)

12x-18-4x+2

16x-20

Thus, the total area of composite figure (circle on a rectangle) is found as  29.91 m².

Area of sector of the circle = (Ф/360°) *π*r²

r is the radius = 5.5 m

π = 3.14

Area =  (30/360°) *3.14 * 5.5²

Area = 7.91 m²

Area of rectangle = width × length

Area = 4 × 5.5 = 22 m²

Total area of composite figure = area of the circle's sector + area of the rectangle

Total area of composite figure =  7.91 m² + 22 m² = 29.91 m²

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After solving the given equations we know that the value of x and y are -4 and -7 respectively.

A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions.

Ax+By=C is the usual form for two-variable linear equations.

A standard form linear equation is, for instance, 2x+3y=5. When an equation is given in this format, finding both intercepts is rather simple (x and y).

This form is also highly useful when solving systems involving two linear equations.

So, we have the equations:

y=2x+1 ...(1)

-4x+y=9 ...(2)

Now, substitute y=2x+1 in equation (2):

-4x+y=9

-4x+2x+1=9

-2x=8

x=-4

Now, insert x=-4 in equation (1):

y=2x+1

y=2(-4)+1

y=-8+1

y=-7

Therefore, after solving the given equations we know that the value of x and y are -4 and -7 respectively.

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The products written in scientific notation are:

a) [tex]5.88*10^9[/tex]

b) [tex]15*10^{12}\\[/tex]

c) [tex]8.4*10^{-10}[/tex]

d) [tex]42*10^{-7}[/tex]

The first product we need to solve is:

[tex](4.2*10^6)*(1.4*10^3)[/tex]

We can reorder that product into the one below:

[tex](4.2*1.4)*(10^6*10^3)[/tex]

The exponents in the right side are added, so we get:

[tex](4.2*1.4)*(10^6*10^3) = 5.88*10^{3 + 6} = 5.88*10^9[/tex]

Now let's do the same in the others.

b)

[tex](5*10^5)*(3*10^7)\\= 5*3*10^{5 + 7}\\= 15*10^{12}\\[/tex]

Now we need to write the first number between 0 and 10, then we can rewrite this as:

[tex]1.5*10^{13}[/tex]

c) Again doing the same thing.

[tex](4*10^{-3})*(2.1*10^{-7}})\\= 4*2.1*10^{-3 - 7}\\= 8.4*10^{-10}[/tex]

d) Finally, this last product is:

[tex](6*10^{-2})*(7*10^{-5}})\\= 6*7*10^{-2 - 5}\\= 42*10^{-7}[/tex]

Again we need to have a number between 0 and 10, so we can rewrite this as:

[tex]4.2*10^{-8}[/tex]

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The letters of the circled points based on their second coordinate values, from least to greatest R, A, T, I, and O.

A word that describes a way to compare quantities is RATIO.

In Mathematics, an ordered pair is sometimes referred to as a coordinate and it can be defined as a pair of two (2) elements that are commonly written in a fixed order within parentheses as (x, y), which represents the x-coordinate (abscissa) and the y-coordinate (ordinate) on the coordinate plane of any graph of a system of linear equations.

In this scenario, we can reasonably infer and logically deduce that the ordered pair representing the circled points are as follows:

Next, we would order the points by their second coordinate values, from least to greatest:

R = -4, A = 1, T = 2, I = 4, O = 5 ⇒ RATIO.

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Answer: D. Yes, the sum of the square of the legs is equal to the square of the hypotenuse.

Step-by-step explanation:

Pythagorean theorem is a^2 + b^2 = c^2

given side length of 6 and 8 and hypotenuse of 10

6^2 + 8^2 = 10^2

36 + 64 = 100

100 = 100

so yes, the sum of the square of the legs = the square of the hypotenuse.

Answer:

True

Step-by-step explanation:

To tell if you can make a triangle or not, it is necessary that the two shorter lengths added up must equal the longer length. In this case, the two shorter lengths are 5 and 5. 5 and 5 added up is 10 and 10>8 meaning that in this case, you can make a triangle

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If the perimeter of a rectangle is 22 inches, then the value of x is 1.

We can use the formula for the perimeter of a rectangle, which is:

[tex]$$P = 2l + 2w$$[/tex]

where P is the perimeter, l is the length, and w is the width.

We are given that the perimeter is 22 inches, so we can substitute P = 22 in the formula and w = 5:

[tex]$$22 = 2l + 2(5)$$[/tex]

Simplifying this equation, we get:

[tex]$$22 = 2l + 10$$[/tex]

[tex]$$2l = 12$$[/tex]

[tex]$$l = 6x$$[/tex]

So the length of the rectangle is 6x inches.

We can now substitute the values of l and w into the formula for the perimeter:

[tex]$$22 = 2(6x) + 2(5)$$[/tex]

Simplifying this equation, we get:

[tex]$$22 = 12x + 10$$[/tex]

[tex]$$12x = 12$$[/tex]

[tex]$$x = 1$$[/tex]

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Question:-

The perimeter of a rectangle is 22 inches. The width of the rectangle is 5 and the length is 2x. What is the value of x?

We can see this by noticing that [tex]6^2 = 36 < 37 < 49 =[/tex] [tex]7^{2}[/tex]. Taking the square root of each expression, we get [tex]\sqrt36 < \sqrt37 < \sqrt49,[/tex] which simplifies to [tex]6 < \sqrt37 < 7[/tex].

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, the square root of a number "x" is represented by the symbol "√x". For example, the square root of 25 is 5, because 5 multiplied by 5 equals 25.

Given by the question.

[tex]\sqrt29[/tex] lies between 5 and 6.

We can estimate this by recognizing that. [tex]5^2 = 25 < 29 < 36 =[/tex] [tex]6^{2}[/tex]. Taking the square root of each expression, we get √25 < √29 < √36, which simplifies to [tex]5 < \sqrt29 < 6[/tex].

[tex]\sqrt37[/tex]lies between 6 and 7.

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The exponential function that models the quail population is:

P(t) = P0 * (1 + r)^t

where:

P0 is the initial population (385)

r is the annual growth rate (30% or 0.3)

t is the time in years

Substituting the values, we get:

P(t) = 385 * (1 + 0.3)^t

Simplifying:

P(t) = 385 * 1.3^t

To find the approximate population after 5 years, we substitute t = 5:

P(5) = 385 * 1.3^5

P(5) = 385 * 3.277

P(5) = 1262.45

Therefore, the approximate population after 5 years is 1262 quail