how to rationalize the denominator with a square root

Step-by-step explanation:

Factor numerator and denominator:

6(x-4)(x+2)   /   3 ( x-6)(x+2)        cancel out x+2    divide 6/3

2(x-4) / (x-6)

Answer:

SHORT SOLUTION STEPS

(6x^2−12x−48)÷(3x^2 −12x−36)

Factor the expressions that are not already factored.

6(x−4)(x+2) / 3(x−6)(x+2)​

Cancel out 3(x+2) in both numerator and denominator.

2(x−4) / x-6

​

Expand the expression.

2x−8 / x−6

Answer= 2x-8 / x-6

Answer:

starting with 6 and adding 3 an hour over two days or 48 hours that  would be  3x48= 144 then add your original 6 that would be 150

Step-by-step explanation:

6 + (3x48) = logs for two days

6 + 144 = 150

150!!!

a. The number of logs needed for one night can be represented by the expression:

6 + 3h

b. The expression for the number of logs needed for 2 nights is 12 + 6h.

A term is a single mathematical phrase. It might consist of only one variable—a letter—one number—positive or negative—or several variables multiplied but never added or subtracted. A number is placed in front of some nouns that have variables. Coefficients are numbers that come before phrases.

If Nelson plans to start the fire with 6 logs and add 3 logs for each hour the fire burns, then the number of logs needed for one night can be represented by the expression:

6 + 3h

where h is the number of hours the fire burns.

To find the number of logs needed for 2 nights, we can simply multiply the number of logs needed for one night by 2:

2(6 + 3h)

Simplifying this expression gives:

12 + 6h

So the expression for the number of logs needed for 2 nights is 12 + 6h. This represents starting the first night's fire with 6 logs, adding 3 logs per hour for that night, and then starting the second night's fire with 6 logs again, and adding 3 logs per hour for that night as well.

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Answer: Mr. Waddle weighs 328 lbs

Step-by-step explanation:

Mr. Waddle weighs 159 lbs aged 28 yrs old in 1988

He weighed 169 lbs aged 31 yrs old in 1991

Yes, the wait time significantly decreased after implementing the new system, based on hypothesis testing.

To determine if the wait time has significantly decreased, we can conduct a hypothesis test using the one-sample t-test. The null hypothesis, denoted as H0, is that the population mean waiting time is still 84.3 seconds, while the alternative hypothesis, denoted as Ha, is that the population mean waiting time has decreased to less than 84.3 seconds.

Let's set the significance level at α = 0.05. Since the sample size is large (n = 45), we can use the t-distribution to conduct the hypothesis test. The test statistic can be calculated as:

t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))

t = (79.1 - 84.3) / (17.3 / sqrt(45))

t = -2.16

Using a t-distribution table or calculator with 44 degrees of freedom

(df = n - 1), the p-value for this test is found to be approximately 0.018.

Since the p-value (0.018) is less than the significance level (0.05), we reject the null hypothesis. This means we have evidence to suggest that the wait time has significantly decreased after implementing the new drive-through system.

Therefore, based on the given data, we can conclude that the wait time has significantly decreased at the fast-food restaurant after implementing the new drive-through system.

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Answer:\(\Large\boxed{First~choice.~y\leq \frac{1}{2}x+2 }\)

Step-by-step explanation:

Given function form

Point-Slope form : y = mx + b

Given the information from the graph

Since the graph crosses (0, 2), b = 2

Find the slope

Choose any two points from the graph

(x₁, y₁) = (0, 2)

(x₂, y₂) = (2, 3)

Slope = (y₂ - y₁) / (x₂ - x₁)

Slope = (3 - 2) / (2 - 0)

Slope = 1 / 2

m = 1 / 2

Therefore, the equation form of the inequality is y = (1/2)x + 2

Since it is a solid line, the values on the line are also included. Thus, it must be either greater than or equal to, or less than or equal to.

The shaded region is below the solid line, therefore, it is ≤

Therefore, the inequality is \(\Large\boxed{y\leq \frac{1}{2}x+2 }\)

To double-check the sign, we can substitute (0, 0) into the inequality to see whether it is true.

y ≤ (1/2) x + 2

0 ≤ (1/2) 0 + 2

0 ≤ 2              TRUE

Hope this helps!! :)

Please let me know if you have any questions

Step-by-step explanation:

What is a computer ?♡?×_×7×^××:×::×:+^++^+_+

Answer:

There are 20 nickels and 12 quarters.

Step-by-step explanation:

20 nickels = 1 dollar.

12 quarters = 3 dollars.

$1 + $3 = $4

Answer:

Step-by-step explanation:

Area of the larger circle:

The shaded area:

Area of the larger circle:

Answer:  C

Step-by-step explanation:

400 multiply by 2 equals 800

and then 800 subtracted by 400 will be 400

and 400 subtracted by 1 will be 399.

Therefore, answer is C

Answer:

4.96 Kilometers

Step-by-step explanation:

Answer:

12.9 kilometers

Step-by-step explanation:

Find how many kilometers Kyle ran by dividing 8 by 0.62:

8/0.62

= 12.9

So, Kyle ran approximately 12.9 kilometers

In problem 7, the given angle measures form a valid triangle. In problems 8, 9, and 10, the given angle measures do not form a valid triangle, and the first angle measure in problem 8 needs to be changed to make it a valid triangle.

The sum of angles in a triangle is 180°, but 76.2° + 81.7° + 22.1° = 180°. Therefore, these angle measures form a triangle.

The sum of the angles in a triangle is 180°, but 115.1° + 47.5° + 93° ≠ 180°. The first angle needs to be changed to make it a valid triangle. Let x be the new measure of the first angle. Then, we have

x + 47.5° + 93° = 180°

x = 39.5°

So, the new angle measures are 39.5°, 47.5°, and 93°, which form a triangle.

The sum of the angles in a triangle is 180°, but 5 2/3° + 61 1/3° + 87° ≠ 180°. We need to convert the mixed numbers to improper fractions and then add them

5 2/3° = 5 + 2/3 = 17/3°

61 1/3° = 61 + 1/3 = 184/3°

17/3° + 184/3° + 87° = 207°

Since 207° is greater than 180°, these angle measures cannot form a triangle.

The sum of the angles in a triangle is 180°, but 31 3/4° + 53 1/2° + 94 3/4° ≠ 180°. We need to convert the mixed numbers to improper fractions and then add them

31 3/4° = 31 + 3/4 = 127/4°

53 1/2° = 53 + 1/2 = 107/2°

94 3/4° = 94 + 3/4 = 379/4°

127/4° + 107/2° + 379/4° = 359/2°

Since 359/2° is greater than 180°, these angle measures cannot form a triangle.

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Answer:

try to copy and paste on gooogle and i think the answer is going to pop up

Step-by-step explanation:

Answer:

Step-by-step explanation:

x>2/3

The length of the street is 579.33 feet.

The distance along a circle's perimeter is referred to as its circumference.

Given:

City workers are using a snowplow to clear a street.

A tire on the snowplow is 4.1 feet in diameter and has to turn 45 times in traveling the length of the street.

The radius of the snowplow is 2.05 feet.

The length of the street = the circumference of the snowplow times 45

The length of the street = 45 x 2 x 3.14 x 2.05

The length of the street = 579.33 feet.

Therefore, The length of the street = 579.33 feet.

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Answer:

standard form: −3x³+5x+6

degree and number of terms.

The degree of a polynomial is the highest degree of its terms. 3

The leading term in a polynomial is the term with the highest degree.

−3x³

The leading coefficient of a polynomial is the coefficient of the leading term. −3

List the results.

Polynomial Degree: 3

Leading Term: −3x³

Leading Coefficient: −3

Step-by-step explanation:

The balance of the unsubsidized loan at the time of graduation is $3,691.10.

Given:

Principal amount = $2,560

Interest rate = 6.8% = 0.068

Time period = 6 months

We can use the formula for compound interest to calculate the future value:

\(Future\; Value = P (1 + \dfrac{Interest Rate}{Number of Periods}) ^{(Periods * Time )\)

Substituting the value back into the formula as

Future Value = \($2,560 (1 + \dfrac{0.068}{12} )^{(12 * 6)\)

                      = \($2,560 (1 + 0.00566667)^{(72)\)

                      = \($2,560 (1.00566667)^{(72)\)

                      = $2,560 * 1.44044291

                       = $3,691.10

Therefore, the balance is $3,691.10.

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i)  The Cobb-Douglas production function y = \(Ax^1h^1x^2b^2\) is negatively sloped and convex to the origin. ii) These positive signs ensure that the partial derivatives and second-order partial derivatives are positive or non-negative. iii) This equation represents the isocline defined by RTS = 1 for the given Cobb-Douglas production function.

i) To prove that the Cobb-Douglas production function y = Ax^1h^1x^2b^2 is negatively sloped and convex to the origin, we need to show that the partial derivatives with respect to x1 and x2 are positive, and the second-order partial derivatives are non-negative.

Partial derivatives:

∂y/∂x1 = A *\((1 * x^1 * h^1 * x^2b^2) / x1 = A * h^1 * x^2b^2\)

∂y/∂x2 = A *\((1 * x^1 * h^1 * x^2b^2) / x2 = A * x^1 * h^1 * b^2 * x2(b^2-1)\)

The partial derivatives are positive since A, h^1, and x^1 are assumed to be positive parameters.

Second-order partial derivatives:

∂^2y/∂x\(1^2\) = \(A * h^1 * x^2b^2 > 0\)

∂^2y/∂x\(2^2\) = \(A * x^1 * h^1 * b^2 * (b^2-1) * x2(b^2-2) > 0\)

The second-order partial derivatives are non-negative since A, \(h^1,\) and \(b^2\) are assumed to be positive parameters.

Therefore, the Cobb-Douglas production function y = \(Ax^1h^1x^2b^2\) is negatively sloped and convex to the origin.

ii) For the function to be well-behaved, the parameters A, \(h^1,\) and \(b^2\) should have the following signs:

- A should be positive, as it represents the overall productivity level of the production function.

-\(h^1\) should be positive, as it represents the elasticity of output with respect to the input factor x1.

- \(b^2\) should be positive, as it represents the elasticity of output with respect to the input factor x2.

These positive signs ensure that the partial derivatives and second-order partial derivatives are positive or non-negative, leading to a well-behaved and meaningful production function.

iii) The marginal rate of technical substitution (RTS) for a Cobb-Douglas production function is defined as the ratio of the marginal product of one input to the marginal product of the other input:

RTS = (∂y/∂x1) / (∂y/∂x2)

From the partial derivatives calculated earlier, we have:

RTS = \((A * h^1 * x^2b^2) / (A * x^1 * h^1 * b^2 * x2(b^2-1))\)

    =\((x^2b^2) / (x^1 * b^2 * x2(b^2-1))\)

    = \((x^2b^2) / (x^1 * x2(b^2-1))\)

To find the isocline defined by RTS = 1, we set RTS equal to 1:

1 =\((x^2b^2) / (x^1 * x2(b^2-1))\)

Simplifying, we get:

\(x2(b^2-1) = x^1 * x^2b^2\)

This equation represents the isocline defined by RTS = 1 for the given Cobb-Douglas production function.

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Answer:

98

Step-by-step explanation:

Add them all up. 60+20= 80+18 = 98.

When a cylinder is intersected by a plane parallel to the bases, the resulting cross-section is a circle.

A cylinder is a solid geometric figure which has two parallel, congruent bases.

The bases are connected by a curved surface called a lateral surface. If a plane is made to intersect a cylinder, the resulting cross-section can be circular or elliptical, depending on how the plane intersects the cylinder.

In this case, if the plane intersects the cylinder parallel to the bases, the resulting cross-section is a circle because any section taken parallel to the base of a cylinder is congruent to the base.

This is true for both right cylinders and oblique cylinders.

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Answer:

<1 = 33

<2 = 90

<3 = 57

Step-by-step explanation:

<3 = 57 because of alternate interior angles

<2 + <3 + 33 = 180

<2 + 57 + 33 = 180

<2 + 90 = 180

<2 = 90

<1 + <2 + 57 = 180

<1 + 90 + 57 = 180

<1 + 147 = 180

<1 = 33

Answer:

11

Step-by-step explanation:

The f(4) using is found using newton's interpolating polynomials of order 4.

function y=CL10_Exercise(part)

%% Input

% part: string for part a,b,c,d

%

%% Output

% y value of the underlying function at x=4

%

%% Write your code here

X=[1,2,3,5,6];

Y=[15,8,5.5,30,52];

x=4;

y=1;

switch part

case 'a'

%% Newton interpolation (Order 4)

a=X;

b=Y;

%x=input('Enter x: ');

[m,n]=size(a);

fx=0;

for i=1:n

%_____________Calculating Dividing Difference_____________________

s=0;

for j=1:i

p=1;

for k=1:i %Denominator part product

if(k~=j)

p=p*(a(j)-a(k));

end

end

s=s+b(j)/p; %summation f(x)/product

end

%_________________________________________________________________

p=1;

for j=1:i-1 %coefficient part of f[...]

p=p.*(x-a(j));

end

fx=fx+s.*p; %Polynomial!

end

y=fx;

case 'b'

%% not-a-knot spline

case 'c'

%% clamped spline

case 'd'

%% Hermite spline

end

end

Hence, the f(4) using is found using newton's interpolating polynomials of order 4.

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Answer:

To three decimal places, The process capability index is 0.167

Step-by-step explanation:

We have to find the process capability index,

Upper specification = 16.5 ounces,

Lowe specification = 15.5 ounces

Mean = 16.0 ounces

Standard Deviation = S = 1 ounce

process capability index = (upper specification - lower specification)/6S

process capability index = (16.5 - 15.5)/6(1)

= 1/6

Process capability index = 1/6 = 0.16666667

To 3 decimal places, we get,

Process capability index = 0.167

We have the expression:

We can simplify it by dividing the numerator by the denominator (division of polynomials):

The answer is: x²+4x+12+29/(x-3)

Answer:

C. Area

hope this helped

Answer:

B i think

Step-by-step explanation:

The formula for the surface area of the box, in terms of the length of one side of the square base (s), is A = s^2 + 4s^2 = 5s^2.

The derivative of the surface area function with respect to s, denoted as dA/ds, gives us the rate of change of the surface area with respect to the length of one side of the base.

1. The surface area of the box consists of the area of the square base and the four equal sides. The area of the square base is s^2, and each side has a length of s. Therefore, the total surface area is given by A = s^2 + 4s^2 = 5s^2.

2. To find the derivative of the surface area function, we differentiate 5s^2 with respect to s using the power rule of differentiation. The power rule states that if we have a function f(x) = cx^n, then the derivative is f'(x) = cnx^(n-1).

  Applying the power rule, we have dA/ds = d(5s^2)/ds = 10s.

3. The derivative, dA/ds = 10s, represents the rate of change of the surface area with respect to the length of one side of the square base. This means that for every unit increase in s, the surface area increases by 10s units.

  The derivative does not provide information about minimizing the amount of material used. To find the dimensions of the box that minimize the amount of material used, we need to set up an optimization problem and solve for the critical points. This involves setting the derivative equal to zero and finding the values of s that satisfy this equation. However, since the problem statement does not provide a specific volume constraint or objective function, we cannot proceed with the optimization process.

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Distance = 1120 km (Answer).

( Down below is the explanation)

Convert the time to hours only.

1 hour 50 minutes = 1 50/ 60 hour

1 hour 50 minutes = 1 5/6

1 hour 50 minutes = 7/6

Find the distance

Distance = Speed x Time

Distance = 960 x 7/6

Distance = 1120 km.

~ I hope this helped, and I would appreciate Brainliest. ♡ ~ ( I request this to all the lengthy answers I give ! )

The amount earned by the bookstore if all of the packages of supplies are sold is given as follows:

$60.

Before obtaining the amount earned, we must obtain the number of packages, with is given by the greatest common factor of the amounts of each item.

The amounts of each item are given as follows:

36, 48 and 24.

Their greatest common factor is obtained factoring them simultaneously by the same prime factors as follows:

36 - 48 - 24|2

18 - 24 - 12|2

9 - 12 - 6|3

3 - 4 - 2

Hence:

gcf(36, 48, 24) = 2² x 3 = 12.

Which is the number of packages. As each package sells for $5, the amount earned is given as follows:

12 x 5 = $60.

Each package is composed by:

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Answer:

682 is the biggest number but if it is talking about the two numbers added then it is 19 and 682

Edit: If there is supposed to be a comma in-between the two pairs of numbers then the answer would be 32,665. I didn't think they could have been commas at first but if they are then 32,665 is the answer.

Step-by-step explanation:

682 is bigger than all the others. 19 + 682 is bigger than all the other paired ones.