From his tent, Mario can see a tree 120m away on a bearing of 056°. He can also see a rock that is due east of his tent and due south of the tree.

a) Sketch and label a diagram showing this information.
b) Hence find the distance from the rock to the tree.

The domain of the function R(x) is (-∞, -4) U (-4, 3) U (3, ∞).

The vertical asymptote is x = -4 and x = 3

The horizontal asymptote is y = 1.

In Mathematics, a domain refers to the set of all real numbers (x-values) for which a particular function is defined.

In Mathematics, the horizontal portion of any graph is used to represent all domain values and they are both read and written from smaller to larger numerical values, which simply means from the left of any graph to the right.

By critically observing the graph shown in the image attached below, we can logically deduce the following domain:

Domain = {0, ∞}.

In conclusion, an asymptote is a line which the graph of any given function approaches but would never cross or touch;

Vertical asymptote is x = -4 and x = 3

Horizontal asymptote is y = 1.

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Complete Question:

Analyze the graph of the function.

[tex]R(x)=\frac{x^2}{x^2+x-12}[/tex]

a. What is the domain of R(x)?

b. What is the vertical and horizontal asymptote?

By adding fractions that represent each of the amounts of cupcakes, we can see that you need 9 cupcakes for you and your friends.

Her we know that you want 3 halves of a cupcake for yourself, 8 halves for your friend, and 7 halves for your other friend.

So we just need to add all of these fractions, to do so, we need to solve the follwing opeartion:

3*(1/2) + 8*(1/2) + 7*(1/2)

3/2 + 8/2 + 7/2

All of these have the same denominator so we can directly add them up:

3/2 + 8/2 + 7/2 = (3 + 8 + 7)/2

(3 + 8 + 7)/2 = 18/2

18/2 = 9

You need 9 cupcakes for you and your friends.

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Complete question:

"I want 3 halves of a cupcake for myself, 8 halves for my friend, and 7 halves for our other friend. How many halves we need in total?"

1. Batman spent $99 this month after starting with $57 and ending with a balance of -$42.

2. The average change in Barbie's position each minute in the scuba diving is -7.3 meters.

3. The change in Ms. Blue's weight after one year of dieting and losing 4 pounds monthly is -48 pounds.

Initial balance in the bank account = $57

Final balance at month-end = $-42

The total amount spent = $99 ($57 + $42)

Thus, when $99 is taken away from $57, the balance is $-42.

Initial position = -2.1 m

Final position = -38.6 m

Change in position = -36.5 m (-38.6 - -2.1)

The number of minutes used to change position = 5 minutes

The average change per minute = -7.3 m (-36.5 ÷ 5)

The quantity of pounds lost per month = 4 pounds

The period of dieting = 1 year = 12 months

The total change in weight after one year of dieting = -48 pounds (-4 x 12)

Thus, these three problems are solved using mathematical operations.

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Given:-

To find:-

Answer:-

In the given right angled triangle, we may use the trigonometric ratios. We can see that the measure of the longest side is 5 which is hypotenuse and the measure of perpendicular needs to be find out .

We may use the ratio of sine here as , we know that in any right angled triangle,

[tex]\implies\sin\theta =\dfrac{p}{h} \\[/tex]

And here , p = 5 and h = x , so on substituting the respective values, we have;

[tex]\implies \sin\theta = \dfrac{x}{5} \\[/tex]

Again here angle is 45° . So , we have;

[tex]\implies \sin45^o =\dfrac{x}{5} \\[/tex]

The measure of sin45° is 1/√2 . so on substituting this we have;

[tex]\implies \dfrac{1}{\sqrt2}=\dfrac{x}{5} \\[/tex]

[tex]\implies x =\dfrac{5}{\sqrt2}\\[/tex]

Value of √2 is approximately 1.414 . So we have;

[tex]\implies x =\dfrac{5}{1.414} \\[/tex]

[tex]\implies \underline{\underline{\red{\quad x = 3.53\quad }}}\\[/tex]

Hence the value of x is 3.53 .

and we are done!

Answer:

The length of side x to the nearest tenth is 3.5.

Step-by-step explanation:

From inspection of the given right triangle, we can see that the interior angles are 45°, 45° and 90°. Therefore, this triangle is a 45-45-90 triangle.

A 45-45-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : 1 : √2.  Therefore, the formula for the ratio of the sides is b: b : b√2 where:

We have been given the hypotenuse, so:

[tex]\implies b\sqrt{2}=5[/tex]

Solve for b:

[tex]\implies \dfrac{b\sqrt{2}}{\sqrt{2}}=\dfrac{5}{\sqrt{2}}[/tex]

[tex]\implies b=\dfrac{5}{\sqrt{2}}[/tex]

The side labelled "x" is one of the sides opposite the 45° angles, so:

[tex]\implies x=b[/tex]

Substitute the found value of b into the equation for x:

[tex]\implies x=\dfrac{5}{\sqrt{2}}[/tex]

[tex]\implies x=3.5355339...[/tex]

[tex]\implies x=3.5\;\sf(nearest\;tenth)[/tex]

Therefore, the length of side x to the nearest tenth is 3.5.

Answer:

To calculate the interest rate, we can use the formula for the present value of an annuity:

PV = PMT x [(1 - (1 + r)^(-n)) / r]

where:

PV = present value

PMT = monthly payment

r = interest rate per period

n = number of periods

In this case, we have:

PV = $12,300 (the sticker price)

PMT = $420

n = 36 (36 months)

Substituting these values into the formula, we get:

12,300 = 420 x [(1 - (1 + r)^(-36)) / r]

Simplifying this equation algebraically is not possible, so we need to use numerical methods to solve for r. One way to do this is to use a financial calculator or spreadsheet program, which can find the interest rate that makes the equation true. Using Excel's RATE function, for example, we get:

=RATE(36,-420,12300)

This gives us an interest rate of approximately 3.33% per month or 39.96% per year.

Therefore, Ella is being offered an interest rate of 3.33% per month or 39.96% per year.

The probability of choosing a student who plays both basketball and baseball is approximately 0.4643.

It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to occur.

We can use the inclusion-exclusion principle to find the number of students who play both basketball and baseball.

Number of students who play both = Number of basketball players + Number of baseball players - Number of students who play neither

                                                           = 11 + 13 - 11

                                                           = 13

Therefore, out of 28 students, 13 play both basketball and baseball.

The probability of choosing a student who plays both sports can be calculated as follows:

P(plays both) = Number of students who play both / Total number of students

                      = 13/28

                      = 0.4643 (rounded to 4 decimal places)

So the probability of choosing a student who plays both basketball and baseball is approximately 0.4643.

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Answer:

  148 cm²

Step-by-step explanation:

You want the surface area of a cuboid with edge lengths 4 cm, 5 cm, and 6 cm.

The formula is shown in your problem statement.

It can be made easier to compute using a little rearranging:

  SA = 2LW +2LH +2WH

  SA = 2(LW +LH +WH)

  SA = 2(LW +H(L +W))

The choice of dimensions for L, W, and H doesn't matter.

  SA = 2(4·6 +5(4+6)) = 2(24 +5(10)) = 2(74) = 148 . . . . cm²

The surface area is 148 cm².

The tangent ratio for angle F is [tex]\frac{4}{3}[/tex] . The solution has been obtained by using trigonometry.

What is trigonometry?

In the area of mathematics known as trigonometry, right-angled triangles, including their sides, angles, and connections, are studied.

We are given a right angled triangle. The sides of the triangle are given as DF is 6, DE is 8 and EF is 10.

This means that the perpendicular is 8 and base is 6.

We know by trigonometry that tan θ is the ratio of triangle's perpendicular and base.

So, from this we get

⇒ tan F = [tex]\frac{8}{6}[/tex]

⇒ tan F = [tex]\frac{4}{3}[/tex]

Hence, the tangent ratio for angle F is [tex]\frac{4}{3}[/tex] .

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Question: What is the tangent ratio for angle F, if DF is 6, DE is 8 and EF is 10?

The derivative of f(x) is (1/2)(9x - sin²(2x))^(-1/2) * (9 - 4sin(2x)cos(2x)).

What is a derivative?

A derivative is a mathematical concept used in calculus to determine the rate at which a function changes with respect to its input variable. It measures the instantaneous rate of change of the function at a specific point, and is calculated as the limit of the rate of change as the input variable approaches zero. The derivative is represented by the notation f'(x) or dy/dx, and can be used to find important information about the function, such as its slope, maxima and minima, and points of inflection.

What are expression?

In mathematics, an expression is a combination of numbers, symbols, and/or variables that are combined in some way using mathematical operations such as addition, subtraction, multiplication, division, or exponentiation.

To find the derivative of f(x) = √[9x - sin²(2x)], we can use the chain rule and the power rule of differentiation:

f(x) = √[9x - sin²(2x)]

f'(x) = (1/2)(9x - sin²(2x))^(-1/2) * (9 - 4sin(2x)cos(2x))

Using the chain rule, we first take the derivative of the expression inside the square root, which is 9x - sin²(2x), and then multiply it by the derivative of the expression inside the square root. The derivative of the expression inside the square root is:

(1/2)(9x - sin²(2x))^(-1/2) * (d/dx)(9x - sin²(2x))

Using the chain rule and the power rule of differentiation, we can find the derivative of 9x - sin²(2x) as:

d/dx (9x - sin²(2x)) = 9 - 4sin(2x)cos(2x)

Substituting this into the expression for f'(x), we get:

f'(x) = (1/2)(9x - sin²(2x))^(-1/2) * (9 - 4sin(2x)cos(2x))

So, the derivative of f(x) is (1/2)(9x - sin²(2x))^(-1/2) * (9 - 4sin(2x)cos(2x)).

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The derivative of f(x) is: f'(x) = (9 - 4sin(2x)cos(2x)) / [2√(9x - sin²(2x))].

Differentiation is a mathematical concept used to calculate the rate at which one quantity changes with respect to another. In calculus, it refers to the process of finding the derivative of a function, which gives the slope of the tangent line at any point on the function.

In the given question,

To find the derivative of f(x), we can use the chain rule and the power rule of differentiation.

f(x) = √[9x - sin²(2x)]

Let's first simplify the expression under the square root:

g(x) = 9x - sin²(2x)

Taking the derivative of g(x) gives:

g'(x) = 9 - 2sin(2x) * cos(2x) * 2

g'(x) = 9 - 4sin(2x)cos(2x)

Now we can use the chain rule to find f'(x):

f'(x) = [1/2(9x - sin²(2x))^(-1/2)] * g'(x)

f'(x) = [1/2√(9x - sin²(2x))] * [9 - 4sin(2x)cos(2x)]

Therefore, the derivative of f(x) is:

f'(x) = (9 - 4sin(2x)cos(2x)) / [2√(9x - sin²(2x))]

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275 miles can be driven in one day on a budget of $100.

an inequality is defines as a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on either sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

The rental car costs $45 a day and $0.20 a mile. So, for a budget of $100, the inequalities can be written as

45+.20m≤ 100

Where m= no.of miles

No.of miles can be travelled in 100

.2m=100-45

m=55/.2

m=275miles

Hence,275 miles can be driven in one day on a budget of $100.

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Answer:

Option C would likely produce the most representative sample of the population of smartphones made by the company. By testing every 500th smartphone off the production line, the manufacturer can ensure that they are selecting a random sample of smartphones from throughout the production process. This can help to account for any variations or changes that may occur in the manufacturing process over time.

Option A, testing the first smartphone off the production line each day, may not be representative of the entire production run since the first smartphone may not be representative of the rest of the batch. Similarly, option B, testing the last 1000 smartphones produced each month, may not be representative of the entire production run as there may be variations in quality and safety standards throughout the month.

Option D, testing every smartphone made on Wednesday, would likely not produce a representative sample as it may not account for any variations or changes that occur during other days of the week.

The survey results, it can be estimated that approximately 17.93% of adult Americans have seen a ghost at some point in their lives[tex] (718/4,005 = 0.1793)[/tex].

However, it is important to note that this estimate is based on a sample and there may be some degree of error or variability.

Additionally, the term "supernatural experiences" may encompass a wide range of phenomena beyond just seeing ghosts, so the estimate may not accurately reflect the prevalence of all types of supernatural experiences in the population.

In the given survey on supernatural experiences, 718 out of 4,005 adult Americans reported having seen a ghost. Assuming this sample is representative of the adult American population, you can calculate the proportion of adults who have seen a ghost.

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Answer:

<AOB = 112

<AOD = 68

<DOC = 112

Hoped this helped

: )

Argon is the gas who would work best for something that needs a steady, nonreactive gas. (Ar).

When a neutral environment is required, argon is frequently used. In this manner, titanium and other volatile metals are produced. Additionally, it is used in the manufacture of incandescent bulbs to prevent air from corroding their filaments and by welders to safeguard the weld region.

A chemical substance in the periodic table's 18th group is argon. It is an expensive gas and the third most common gas in the atmosphere of the planet. Aside from nitrogen and oxygen, argon is the most prevalent gas in the atmosphere.

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The sum of the measures of the two angles is 180 degrees, as they are angles of a straight line. The measure of the first angle is 4x = 72 degrees, and the measure of the second angle is 6x = 108 degrees.

What is angle ?

A measure of rotation of two crossing lines or planes in mathematics is called an angle. Angles are frequently defined as degrees or radians. Two lines or splines intersect at a location, known as the apex of the angle, to produce an angle. The edges of the angle are the two lines and line segments. According to its measurement, an angle can be categorized as: An acute angle is one that ranges from 0 to 90 degrees. Right arc: an angle that is 90 degrees in length. A measureable angle intermediate 90 and 180 degrees is referred to as an obtuse angle. 180 degree angle is referred to as a straight angle.

The sum of the measures of the two angles is 180 degrees, because they are angles of a straight line. Therefore:

4x + 6x = 180

Simplifying the left-hand side, we get:

10x = 180

Dividing both sides by 10, we obtain:

x = 18

So the measure of the first angle is:

4x = 4(18) = 72 degrees

And the measure of the second angle is:

6x = 6(18) = 108 degrees

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The complete question is: What is the measurement of an angle if its measure is 4x and another angle's measure is 6x?

The chain rule is used when a function is composed of two or more functions, where one function is applied to the output of the other. In other words, it is used when a function is "nested" inside another function. Thus, option A is correct.

We can solve this problem by applying the chain rule and the product rule of differentiation. Let's begin by simplifying the expression inside the parentheses:

[tex]y = ex (e^-2x + 7)[/tex]

Now, let's apply the product rule:

[tex]dy/dx = (ex)'(e^-2x + 7) + ex(e^-2x + 7)'[/tex]

Using the chain rule, we can find (ex)':

[tex](ex)' = ex[/tex]

And using the chain rule again, we can find [tex](e^-2x + 7)':[/tex]

[tex](e^-2x + 7)' = (-2e^-2x)[/tex]

Substituting these values back into the original equation, we get:

[tex]dy/dx = ex(e^-2x + 7) + ex(-2e^-2x)[/tex]

Simplifying, we get:

[tex]dy/dx = ex(e^-2x + 7 - 2e^-2x)[/tex]

[tex]dy/dx = ex(e^-2x - e^-2x + 7)[/tex]

[tex]dy/dx = ex(7)[/tex]

Therefore, the answer is A) 7.

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Answer:

the answer is f-1(x) = 5 - ln (x - 6)/ln (3) (I guess?)

Answer:

To determine which statements are true, we can use the standard form of the equation of a circle:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius.

Using this form, we can rewrite the given equation as:

(x - 1)^2 + y^2 = 3^2 + 1^2 = 10

Comparing this to the standard form, we can see that the center of the circle is (1, 0), so the statement "The center of the circle lies on the x-axis" is true. However, the statement "The center of the circle lies on the y-axis" is false.

To find the radius, we can rearrange the equation as follows:

x^2 - 2x + y^2 = 8

Completing the square for x, we get:

(x - 1)^2 + y^2 = 9

This shows that the radius of the circle is 3, so the statement "The radius of the circle is 3 units" is true, as well as the statement "The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9."

Therefore, the three true statements are:

1.The radius of the circle is 3 units.

2.The center of the circle lies on the x-axis.

3.The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Step-by-step explanation:

hope its help <:

The compound interest rate of the given information for which the annual rate of the bank account is = 3.9%.

What about compound interest?

Compound interest refers to the interest that is earned not only on the principal amount of a loan or investment but also on any interest that has been previously earned. In other words, it is the interest calculated on the initial principal and on the accumulated interest of previous periods. This compounding effect can result in significant growth over time, as the interest earned in each period is added to the principal, and the interest earned in subsequent periods is calculated on the new, larger principal amount. Compound interest is often used in long-term savings and investment accounts, such as retirement funds or fixed deposits, as it can result in a higher return than simple interest over time.

According to the given information:

In this case we know that;

⇒ A = [tex]P(1 + r)^n[/tex]

⇒ A = amount

⇒ P = principal

⇒ r = rate

⇒ n = number of years

⇒ 710.89  = 658.20[tex](1 + r)^2[/tex]

⇒ 710.89/658.20 = [tex](1 + r)^2[/tex]

⇒ 1.08 = [tex](1 + r)^2[/tex]

⇒ ln(1.08) =ln [tex](1 + r)^2[/tex]

⇒ 0.077 = 2ln[tex](1 + r)[/tex]

⇒ 0.077/2 = ln(1 + r)

⇒ ln[tex](1 + r)[/tex] = 0.0385

⇒ 1 + r = 1.039

⇒ r = 1.039 - 1

⇒ r = 0.039

⇒ r = 3.9%

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Altitude is in a polygon, a perpendicular segment from a vertex to the opposite side or to the line containing the opposite side.

In geometry, the hypotenuse is the longest side of a right-angled triangle, opposite to the right angle. It is also the side that connects the two other sides, which are called the adjacent and opposite sides.

One of the key properties of the geometric mean is that it is always less than or equal to the arithmetic mean (the regular average) of the same set of numbers, except when all the numbers are equal.

The geometric mean is used in various fields such as finance, economics, biology, and physics. It is particularly useful in situations where values are subject to compounding or exponential growth, and where small changes in values can have a significant impact over time.

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Answer:  36

Step-by-step explanation:

2root3 * 3root12

2*3*root3*root12

6root36

root36=6

6*6=36

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By circle , 15 is the length of x .

What is a circle, exactly?

A circle is a closed, two-dimensional object where every point in the plane is equally spaced from a central point. The line of reflection symmetry is formed by each line that traverses the circle.

                                Moreover, every angle has rotational symmetry around the center. With no sides or edges, a circle is a figure with a round shape. A circle can be characterized in geometry as a closed shape, a two-dimensional shape, or a curved shape.

AE² = EC . CD

10² = 5 * x

100 = 5 * x

100/5 = x

 20 = x

ED = 20 - 5   = 15

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Using ratios, we can find the average weight of the adult bear to be 250 pounds and the angles of the triangles are 10°, 75° and 95°.

Comparing two amounts of the same unit and calculating the ratio allows us to determine how much of one quantity is included in the other. Two categories of ratios exist. The part to whole ratio is one, and the part-to-part ratio is another. The part-to-part ratio illustrates the relationship between two separate entities or groupings.

In the question given,

Ratio between birth weight and adult weight of a bear is 3:1000.

The average birth weight as given here is = 12 ounces.

Now let the average adult weight be = x

According to the ratio,

12/x = 3/1000

⇒ 3x = 12000

⇒ x = 4000 ounces.

Now 16 ounces = 1 pounds

4000 ounces

= 4000/16

= 250 pounds.

The angles are in the ratio 2:15:19

Now we know all the angles sum up to 180°

Let the angle be = x.

The equation is as follows:

2x + 15x + 19x = 180

⇒ 36x = 180

⇒ x = 180/36

⇒ x = 5

So, all the angles of the triangles are, 10°, 75° and 95°.

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A square is not created when a plane slices through a cone.

What is square?

A square is a geometrical shape that has four equal sides and four equal angles of 90 degrees.

What is cone?

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a pointed top or apex. It looks like a funnel or an ice cream cone.

When a plane intersects or slices through a cone, it creates a variety of different shapes depending on the angle of the plane and the position of the cone.

If the plane intersects the cone at an angle perpendicular to its base, it will create a circle. If the plane intersects the cone at an angle that is not perpendicular to its base, it will create an ellipse.

If the plane intersects the cone at an angle that is parallel to one of its generators (the lines that can be drawn from the vertex of the cone to any point on its base), it will create a parabola. If the plane intersects the cone at an angle that is not parallel to one of its generators, it will create a hyperbola.

Therefore, a square is not created when a plane slices through a cone.

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The amounts of each substance needed to obtain a mixture of 20 pounds at a price of $5.4 per pound is given as follows:

The amounts are obtained by a system of equations, for which the variables are given as follows:

The mixture has a total of 20 pounds, hence:

x + y = 20.

y = 20 - x.

The total cost of the mixture is of $108, as 108/20 = 5.4, hence:

7x + 3y = 108

Replacing the second equation into the first, the value of x is given as follows:

7x + 3(20 - x) = 108

4x = 48

x = 12.

Then the value of y is given as follows:

y = 20 - 12

y = 8.

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Answer: the missing digit would be 2

721,846 rounded to the nearest ten thousand is 720,000

The length of the rectangle is 10.5 units.

What is perimeter?

Perimeter is the distance of a two-dimensional shape. It is equal to the sum of the lengths of all the sides of the shape. The perimeter is measured in units, such as centimeters, meters, feet, or inches, depending on the unit of measurement used for the dimensions of the shape.

The perimeter of a rectangle is given by the formula:

P = 2l + 2w

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

In this case, we know that the perimeter is 34 units, and the width is 6.5 units. So we can substitute these values into the formula and solve for the length:

34 = 2l + 2(6.5)

Simplifying, we get:

34 = 2l + 13

Subtracting 13 from both sides, we get:

21 = 2l

Dividing both sides by 2, we get:

l = 10.5

So the equation to determine the length of the rectangle is:

2l + 2w = P

Substituting the known values, we get:

2l + 2(6.5) = 34

Simplifying and solving for l, we get:

2l + 13 = 34

2l = 21

l = 10.5

Therefore, the length of the rectangle is 10.5 units.

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Complete question: The perimeter of a rectangle is 34 units, its width is 6.5 units. Write an equation to determine the length (l) of the rectangle.

The fraction that is equivalent to 5/7 is 15/21

It is important to note that fractions are simply described as part of a whole number or element.

Also, equivalent expressions are defined as expressions that have the same solution but differ in the mode of arrangement of the values.

From the information given, we have;

The numerator be x

The denominator is 9 less than twice the numerator

This is represented as;

x/2x - 9 = 5/7

Cross multiply the values, we have;

7(x) = 5(2x - 9)

expand the bracket

7x = 10x - 45

collect the like terms, we have;

7x - 10x = -45

-3x = -45

Make 'x' the subject

x = 15

Then, the fraction = 15/21

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The probability that the cupcake picked at random contains walnuts is given as follows:

0.4 = 40%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

There are 30 cupcakes in a tin, hence the total number of outcomes is given as follows:

30.

The number of cupcakes with walnuts is given as follows:

Hence the probability that the cupcake picked at random contains walnuts is obtained as follows:

p = (3 + 9)/30

p = 12/30

p = 0.4.

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The length of the base is 6 inches.

What is triangle?

A triangle is a three-sided polygon, which is a flat shape with straight sides. It is a basic shape in geometry and has many interesting properties that are useful in mathematics and other fields.

The formula for the area of a triangle is A = 1/2 * b * h, where A is the area, b is the length of the base, and h is the height of the triangle.

In this case, we know that the area of the triangle is 21 in² and the height is 7 in. So we can substitute these values into the formula and solve for the base:

21 = 1/2 * b * 7

Multiplying both sides by 2:

42 = b * 7

Dividing both sides by 7:

6 = b

Therefore, the length of the base is 6 inches.

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Complete question : The area of the triangle is 21 in² ad the height is 7 in. what is the length of the base?