simplify:




...............

Answer:

given rational function is undefined for x = 2

Step-by-step explanation:

The given rational function is

[tex]\frac{x^2-4}{2-x}[/tex]

The rational function is undefined if the denominator is zero.

Hence, the given rational function is undefined if

[tex]2-x=0[/tex]

Solve the equation for x

[tex]x=2[/tex]

Therefore, the given rational function is undefined for x = 2

Final answer:

The ratio of x to y, given the equation 3x = 8y, is 8:3. This is found by dividing both sides of the equation by 3y and simplifying.

Explanation:

To find the ratio of x to y, given the equation 3x = 8y, you can divide both sides of the equation by 3 to solve for x in terms of y, or vice versa.

Dividing both sides by 3y yields:

x/y = (8y)/(3y)

Assuming y is not zero, the y terms cancel out on the right side of the equation, giving us:

x/y = 8/3

Method 2: Using fraction notation:

Rewrite the equation with x isolated:

x = (8/3) × y

This also shows that the ratio of x to y is 8:3.

Both methods lead to the same answer: x : y = 8 : 3.

Thus, the ratio of x to y is 8:3.

Reflection symmetry is a symmetry with respect to the reflection and is also know as mirror symmetry, line symmetry or mirror-image symmetry.

When a figure undergoes a reflection and does not change , then the figure is said to has the reflectional symmetry.

An object with reflectional symmetry can be created by reflecting it about an axis called the Line of Symmetry.

Hope this helps ..!!

Thank you :)

The linear function f (x) hat the agency could use to determine its daily rental charges will be;

⇒ f (x) = 0.2x + 20

Where, f (x) is in dollars and x is in miles.

What is linear expression?

A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power.

Given that;

Heidi rented a car for one day and paid $58 for 190 miles.

And, On another day she paid $86 for 330 miles.

Now,

We find the linear function for f (x), where f (x) is in dollars and x is in miles as;

Here the two situations are given for miles and dollars as;

(190, 58) and (330, 86)

Thus, The linear function with two points (190, 58) and (330, 86) is defined as;

⇒ y - y₁ = (y₂ - y₁)/(x₂ - x₁) (x - x₁)

⇒ y - 58 = (86 - 58) / (330 - 190) (x - 190)

⇒ y - 58 = 28 / 140 (x - 190)

⇒ y - 58 = 1/5 (x - 190)

⇒ y - 58 = 0.2 (x - 190)

⇒ y - 58 = 0.2x - 38

⇒ y = 0.2x - 38 + 58

⇒ y = 0.2x + 20

⇒ f (x) = 0.2x + 20

Thus, The linear function f (x) hat the agency could use to determine its daily rental charges will be;

⇒ f (x) = 0.2x + 20

Where, f (x) is in dollars and x is in miles.

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To determine the daily rental charges at a car rental agency based on fixed and per mile fees, the linear function f(x) is found to be $40 + $0.15 per mile.

Linear function f(x):

To solve a linear equation in two variables, at least two equations are required. There are 60 lemonade and 40 orange juice cups.

A linear equation in two variable can be written as ax + by = 0 where a and b are not equal to zero. It is used to represent a straight line.

Given that, rate of lemonade per cup = $2.

And,  rate of orange juice per cup = $3.

Total number of cups sold = 100.

Total earning = $240.

Suppose the number of lemonade cup be x,

And, the number of orange juice cup be y.

Then, as per the question two equations can be formed given as,

x + y = 100 -------------------------(1)

2x + 3y = 240  -------------------------(2)

To find x and y, multiply equation (1) by 2 and substract it from equation (2),

2x + 3y - 2(x + y) = 240 - 2×100

=> y = 40

Substitute y =40 in equation (1) to get x as,

x + 40 = 100

=> x = 60.

Hence the number of lemonade cups are 60 and orange juice cups are 40.

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To find the value of x in the triangle, set up an equation using the triangle angle sum theorem and solve for x. The value of x is 21 degrees.

The problem involves finding the value of x in a triangle with given angle measures. According to the triangle angle sum theorem, the sum of the angles in any triangle is always 180 degrees. In this case, we are given two expressions for the angles in terms of x: x - 4 degrees and 3x degrees, plus a given angle of 100 degrees.

To solve for x, set up an equation using the given angles:

x - 4 + 3x + 100 = 180

Combine like terms to solve for x:

4x - 4 + 100 = 180

4x + 96 = 180

4x = 180 - 96

4x = 84

x = 84 / 4

x = 21

The error in Larry's construction is:

C.  He set the width of the compass to the diameter of the circle.

The steps for the construction of an inscribed equilateral triangle is as follows:

Hence, the error he that he set the width of compass  equal to the diameter of circle.

Instead he should have set the width of the compass equal to the radius of the circle.

The net change of temperatures between the planets is required.

Neptune to Mars is [tex]154^{\circ}\text{C}[/tex]

Earth to Neptune is [tex]228^{\circ}\text{C}[/tex]

Venus to Earth is [tex]446^{\circ}\text{C}[/tex]

The net change value is the distance of the net change from zero.

So, the absolute value of the change will be the net change.

Difference in temperature of Neptune and Mars

[tex]|-214-(-60)|=154^{\circ}\text{C}[/tex]

Difference in temperature of Neptune and Earth

[tex]|-214-14|=228^{\circ}\text{C}[/tex]

Difference in temperature of Venus and Earth

[tex]|460-14|=446^{\circ}\text{C}[/tex]

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Answer:

154 for neptune to mars is correct. The 228 and 446 is wrong.

Step-by-step explanation:.

When a horizontal plane passes through the center of a sphere with a radius of 26 inches, the cross-section formed is a circle with a radius of 26 inches.

let's break down the problem into two parts:

Describe the cross-section: We'll describe the shape formed when a horizontal plane passes through the center of the sphere.

Calculate the dimensions of the cross-section: We'll calculate the dimensions of the cross-section, such as its radius or diameter.

1. Describe the cross-section:

When a horizontal plane passes through the center of a sphere, it divides the sphere into two equal halves. The cross-section formed by this plane and the sphere will be a circle. Since the plane passes through the center of the sphere, the circle will be perfectly centered.

2. Calculate the dimensions of the cross-section:

Since the radius of the sphere is given as 26 inches, the radius of the cross-section (which is also the radius of the circle formed) will also be 26 inches.

So, the cross-section formed by the plane and the sphere is a circle with a radius of 26 inches.

Answer:

its b

Step-by-step explanation:

Answer:

D 7.2 hours; the number of hours students watch television in a week when they do not participate in any outdoor sports

Step-by-step explanation:

Given is a scatter plot which shows the relationship  betweeen the number of hours students spend watching television and the numer of hours they spend playing outdoor sports each week.

From the graph showing the regression line, the regression line has y intercept as 7.2

(When x=0, y =7.2)

Hence correct answer is

D 7.2 hours; the number of hours students watch television in a week when they do not participate in any outdoor sports

Answer: There are 28.375 grams in an ounce.

Step-by-step explanation:

Given: Total grams in a pound =454 grams

Total ounces in a pound = 16 ounces

Therefore, 16 ounces = 454 grams

To find grams in an ounce, we need to divide 454 grams by 16, we get

The total grams in an ounce=[tex]\frac{454}{16}=28.375\ grams[/tex]

∴The total grams in an ounce= 28.375 grams

Hence, there are 28.375 grams in an ounce.

28.375 grams in an ounce.

To calculate how many grams are in an ounce, divide 454 (grams in a pound) by 16 (ounces in a pound) to get 28.375 grams in an ounce.

To find out how many grams are in an ounce:

Answer:

The wedding planner must have purchased 28 small lanterns and 12 large lanterns.

Step-by-step explanation:

Let the number of small and large lanterns purchased be S and L respectively.

S + L = 40  ........ Equation 1

L = 40 - S

25S + 40L = 1180  ....... Equation 2

25S + 40(40 - S) = 1180

25S + 1600 - 40S = 1180

-15S = 1180 - 1600

-15S = -420

S = 28

L = 40 - S

L = 40 - 28

L = 12

The wedding planner must have purchased 28 small lanterns and 12 large lanterns.

Answer

Maximum area A_max = 11250 ft^2

Step-by-Step Explanation

Declaring Variables:-

The length of the rectangle = y

The width of the rectangle = x

Solution:-

- The perimeter of a rectangle can be expressed using the above two variables as follows:

                             Perimeter (P) = 2*Length + 2*Width

                                                     = 2* ( x + y )

- Since the barn is used as one of the sides (let's say y) we can subtract y

we don't need fencing for this side. The length of the fence required L is:

                            Length (L) = P - y

                                                = 2*x + y

- We are given 300 feet of fencing. So we equate the length equal to 300 and develop a linear relationship between width and length of the barn.

                              300 = 2x + y

                              y = 300 - 2x

- The area (A) of the rectangle is given by the following expression:

                              A = Length*width

                              A = x*y

- Substitute the relationship developed between x and y in the Area (A) expression above. Then we have:

                              A = x*(300 - 2x)

                              A = 300x - 2x^2

- We will take first derivative of the expression of area (A) developed with respect to x and find the critical point of the area function by setting the first derivative A'(x) = 0.

                             A(x) = 300x - 2x^2

                             A'(x) = 300 - 4x

                             0 = 300-4x

                             x = 300 / 4 = 75 ft

- The critical point of the given function lies for the width (x) of 75 ft. We will plug in the critical value x = 75 ft back into the original function of Area and find the maximum area.

                             A(x) = 300x - 2x^2

                             A(75) = 300 (75) - 2(75)^2

                             A_max = A(75) = 11250 ft^2

- The maximum area that can be enclosed by the fencing is 11250 ft²

Answer:11250

Step-by-step explanation:

Answer:

Khan Academy said 45 hope it helps! :)

Answer:

All values of x make the equation true. It is an infinite solution.

Answer:

Option D -8.3 cm, 5.8 cm

Step-by-step explanation:

Given : An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 5 cm long. A second side of the triangle is 6.9 cm long.

To find : The longest and shortest possible lengths of the third side of the triangle?

Solution :

First we create the image of the question,

Refer the attached figure below.

Let a triangle ABC , where angle A has a bisector AD such that D is on the side BC.

The theorem is stated for angle bisector is

"Each angle bisector of a triangle divides the opposite side into segments proportional in length to the adjacent sides".

So, according to question,

Let BD=6 cm, DC=5 cm, AB=6.9 cm

and we have to find AC.

Applying the theorem,

[tex]\frac{BD}{DC}=\frac{AB}{AC}[/tex]

[tex]\frac{6}{5}=\frac{6.9}{AC}[/tex]

[tex]AC=\frac{6.9\times 5}{6}[/tex]

[tex]AC=\frac{34.5}{6}[/tex]

[tex]AC=5.75[/tex]

[tex]AC=5.8 cm[/tex]

If we let AC=6.9 cm, find AB

Then,

[tex]\frac{BD}{DC}=\frac{AB}{AC}[/tex]

[tex]\frac{6}{5}=\frac{AB}{6.9}[/tex]

[tex]AB=\frac{6.9\times 6}{5}[/tex]

[tex]AB=\frac{41.4}{6}[/tex]

[tex]AB=8.28[/tex]

[tex]AB=8.3 cm[/tex]

Therefore, The shortest possible length is 5.8 cm and longest possible is 8.3 cm.

Hence, Option D is correct.