One number is 8 less than twice a second number. Find a pair of such numbers so that their product is as small as possible. These two numbers are ____. (Use a comma to separate your numbers.)
The smallest possible product is ____.

Answer: y = x + 8

Step-by-step explanation:

Answer:

Q1    Value of  angle B = 50°

Q2   BC =  1.3

Q3    AB = 2.57

Step-by-step explanation:

For Part 1
The three angles of  a triangle must add up to 180°

Therefore m∠B + 30 + 100 = 180

m∠B + 130 = 180

m∠B = 180 - 30 = 50°

part 2
The law of sines states that, in  a triangle, the ratio of each side to the sine of the angle opposite that side must be the same for all sides

We have side AC opposite ∠B

AC = 2 and we found that m∠B = 50° from part 1

The side BC is opposite ∠A which is 30°

Therefore, applying the law of sines
[tex]\dfrac{{AC}}{\sin 50^\circ} = \dfrac{{BC}}{\sin 30^\circ}\\\\\\\dfrac{2}{\sin 50^\circ} = \dfrac{{BC}}{\sin 30^\circ}\\\\[/tex]

Multiplying both sides by sin 30° we get
[tex]\dfrac{2}{\sin 50^\circ} \times \sin 30^\circ =BC\\\\or\\\\BC = \dfrac{2}{\sin 50^\circ} \times \sin 30^\circ[/tex]

Using a calculator to compute the right side we get
BC = 1.30540 ≈ 1.3

part 3

Similarly
[tex]\dfrac{AB}{\sin 100} = \dfrac{2}{\sin50}\\\\AB = \sin 100 \times \dfrac{2}{\sin 50} = 2.57115 \approx 2.57[/tex]

Solve for the first variable in one of the equations, then substitute the result into the other equation.

Answer:

Point Form:

(−8,−17)

Equation Form: x=−8,y=−17

Answer:

x=-4, y=5

Step-by-step explanation:

Because y=-3x-7 and y=x+9, so y=y, and -3x-7=x+9

-3x-7=x+9,

so 4x=-16

      x=-4

so we put x=-4 to y=-3x-7

      y=(-3)* (-4)-7

      y=12-7

       y=5

1 meter = 1000 millimeter

Answer: :)

Step-by-step explanation:

The metric system is based on the International System of Units (SI), which is a standardized system of measurement used worldwide. The SI unit for length is the meter, and all other units of length in the metric system are derived from it.

The relationship between millimeters and meters can be expressed mathematically as follows:

1 mm = 0.001 m

1 m = 1000 mm

This means that if you have a length measurement in millimeters, you can convert it to meters by dividing by 1000. Conversely, if you have a length measurement in meters, you can convert it to millimeters by multiplying by 1000.

For example, if you have a piece of wire that measures 500 mm long, you can convert this to meters as follows:

500 mm ÷ 1000 = 0.5 m

Similarly, if you have a piece of fabric that measures 2.5 m long, you can convert this to millimeters as follows:

2.5 m x 1000 = 2500 mm

In summary, the relationship between millimeters and meters is that they are both units of length in the metric system, with one meter being equal to 1000 millimeters.

The rational expression is undefined when the denominator is equal to zero. This is because division by zero is not allowed in mathematics, and the result would be "undefined."

To find the values for which the rational expression is undefined, we need to set the denominator equal to zero and solve for the variable. In this case, the denominator is z:

z = 0

Therefore, the rational expression is undefined when z = 0.

So, the values for which the rational expression ((8x)/(z)) is undefined are:

z = 0

I hope this helps! Let me know if you have any further questions.

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Nicole has 8 nickels and 7 dimes.

A system of equations is simply two or more equations that share the same variables.

Let's use a system of equations to solve this problem:

Let x be the number of nickels, and y be the number of dimes.

From the problem, we know that:

x + y = 15 (equation 1) (the total number of coins is 15)

0.05x + 0.1y = 1.1 (equation 2) (the total value of the coins is $1.10)

To solve for x and y, we can use substitution or elimination. Let's use elimination:

Multiply equation 1 by 0.1:

0.1x + 0.1y = 1.5 (equation 3)

Subtract equation 3 from equation 2:

0.05x = 0.4

x = 8

Substitute x = 8 into equation 1:

8 + y = 15

y = 7

Therefore, Nicole has 8 nickels and 7 dimes.

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The new function after the parent function f(x)=2^(x) is horizontally shrunk by a factor of two, translated right by 10 and translated 5 units down is f(x) = 2^((x-10)/2) - 5.

- Horizontal shrink by a factor of two: This means that the x-value of each point on the graph will be divided by 2. To achieve this, we can multiply the x-value inside the function by 1/2, or divide it by 2. So, the function becomes

f(x) = 2^((x/2))

- Translation right by 10: This means that the entire graph will shift 10 units to the right. To achieve this, we can subtract 10 from the x-value inside the function. So, the function becomes

f(x) = 2^((x-10)/2)

- Translation 5 units down: This means that the entire graph will shift 5 units down. To achieve this, we can subtract 5 from the entire function. So, the function becomes

f(x) = 2^((x-10)/2) - 5

Therefore, the new function is f(x) = 2^((x-10)/2) - 5

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Answer:

See explanation below

Step-by-step explanation:

We have RO parallel to LF

Segments RF and LO are transversal segments

For two parallel lines intersected by a transversal , the alternate interior angles are equal

In this figure, there are two pairs of alternate interior angles

∠O and ∠L form one pair and are equal
∠R and ∠F form another pair and are equal

Since the point T is formed by the intersection of two straight line segments,  the vertically opposite angles must be equal
m∠RTO = m∠LTF

So in the two triangles, ΔRTO and Δ FTL we have

m∠R = m∠F
m∠O = m∠L
m∠RTO = m∠LTF

By the AAA theorem of similarity of angles the two triangles are similar

AAA Similarity Criterion for Two Triangles

The Angle-Angle-Angle (AAA) criterion for the similarity of triangles states that “If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar”.

The set builder nοtatiοn wοuld be {(x, y) | x = 1/125, 1/25, 1/5, 1, 5, 25, 125 and y = -3, -2, -1, 0, 1, 2, 3}

A set builder nοtatiοn is a cοncise way οf representing a set οf elements based οn a rule οr cοnditiοn. In set builder nοtatiοn, we use braces { } tο enclοse the set οf elements, and a vertical bar | οr a cοlοn : tο separate the cοnditiοn frοm the elements. The general fοrmat οf a set builder nοtatiοn is:

{elements | cοnditiοn}

Using this nοtatiοn, we can represent the set οf οrdered pairs (x, y) given in the questiοn as:

{(x, y) | x = 1/125, 1/25, 1/5, 1, 5, 25, 125 and y = -3, -2, -1, 0, 1, 2, 3}

This nοtatiοn reads as "the set οf all οrdered pairs (x, y) such that x takes οn the values 1/125, 1/25, 1/5, 1, 5, 25, and 125, and y takes οn the values -3, -2, -1, 0, 1, 2, and 3."

Hence, the set builder nοtatiοn wοuld be {(x, y) | x = 1/125, 1/25, 1/5, 1, 5, 25, 125 and y = -3, -2, -1, 0, 1, 2, 3}

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An equation that could describe the graph below include the following: A. y = (1/2)^x + 6.

In Mathematics, an exponential function can be modeled by using this mathematical expression:

f(x) = a(b)^x

Where:

Generally speaking, When the base value or y-intercept (a) is less than one (1), the graph of an exponential function increases exponentially to the left. Additionally, the smaller the value of y-intercept (a), the steeper the slope (b) of the line.

By critically observing the graph of this exponential function, we can logically deduce that it was vertically shifted up by 6 units.

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In a linear equation, There is one more test this quarter 39.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                    Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables. Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

lowest average score = 80% * 50  = 40

 suppose the final exam score she must get at least x.

   ( x + 45 + 38 + 36) = 4 ≥ 40

                                x ≥ 39

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Answer:

∠M = ∠P = 112

∠N = ∠Q = 68

Step-by-step explanation:

The interior angles of a quadrilateral always sum up to 360°.

since NP//MP and NM//PQ => ∠N = ∠Q and ∠M = ∠P

and ∠N + ∠Q + ∠M + ∠P = 360

∠M = ∠P

6x - 2 = 4x + 36

6x - 4x = 36 + 2

2x = 38

x = 38/2 = 19

=> ∠M = ∠P = 4(19) + 36 = 112

∠N + ∠Q + ∠M + ∠P = 360

2∠N + 2∠M = 360

∠N + ∠M = 180

∠N = 180 - ∠M = 180 - 112 = 68

∠N = ∠Q = 68

Yes, the binomial 2x+3 is a factor of the given polynomial P(x)=2x^(3)+3x^(2)-2x-3. We can verify this by using synthetic division.

First, we need to find the zero of the binomial, which is -3/2. Then, we can use synthetic division to divide the polynomial by the binomial.

Here are the steps for synthetic division:

1. Write the coefficients of the polynomial in a row: 2  3  -2  -3
2. Write the zero of the binomial to the left of the row: -3/2 | 2  3  -2  -3
3. Bring down the first coefficient: -3/2 | 2  3  -2  -3
               0  -3   6  -6
               ---------------
               2  0   4   -9
4. Multiply the zero by the first coefficient and write the result below the second coefficient: -3/2 * 2 = -3
5. Add the second coefficient and the result: 3 + (-3) = 0
6. Repeat steps 4 and 5 for the remaining coefficients: -3/2 * 0 = 0, -2 + 0 = -2; -3/2 * -2 = 3, -3 + 3 = 0
7. The final row of numbers represents the coefficients of the quotient: 2x^(2) + 0x + 4, with a remainder of -9.

Since the remainder is not zero, the binomial is not a factor of the polynomial. However, if we divide the polynomial by the binomial using long division, we get the following:

2x^(2) + 4 - (9/(2x+3))

This means that the polynomial can be written as: P(x) = (2x+3)(2x^(2) + 4) - 9

So, the binomial 2x+3 is a factor of the polynomial P(x)=2x^(3)+3x^(2)-2x-3.

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Answer:

Below

Step-by-step explanation:

The graph of the linear inequality y ≥ −2x − 1 is the region _______ the graph of the line y= - 2x -1

≥  is greater than or equal to and translates to 'on or above'

The area of the rectangle is 56 square yards.

What is the area of rectangle?

The area of a rectangle is the amount of space or surface inside the shape, measured in square units. It is calculated by multiplying the length of the rectangle by its width.

The formula for the area of a rectangle is:

Area = length x width

In this case, the width is 7 yards and the length is 8 yards. Therefore, the area of the rectangle is:

Area = 7 yards x 8 yards = 56 square yards

So the area of the rectangle is 56 square yards.

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Answer:

Dude.....it's a CIRCLE!!

The provide situation can be modeled as a linear equation, y = 50 + 20x, where y is total cost and x is number of person who will attend the party. The slope of line tells me that the additional cost per person attending the party is $20. Thus, write option is option (b).

Emmett wants to through a party at an escape room. There are two charges for

escape room, one is booking fee and other one is additional cost per person attending the party. Let the booking fee and additional cost per person who attended the party of escape room be 'a ' and 'b'. Also, consider total person who will attend the party are equal to 'x'. So, total cost = booking cost + additional cost per person × number of people's who will attend the party. Let total cost of escape room for x people be 'y'. Then,

y = a + bx --(1)

it is act as modeled linear relationship, see above graph, which represents it graphically. In the graph initial total cost is 50, when no additional cost occur, that is booking fee equals to $50 or a = $50. After that, number of people increase total cost also increases. When there are 10 people in party then total cost is $250 and the additional cost = $200 for 10 people. So, additional cost per person

= 200/10 = 20 , i.e., b = $20 and equation(1) is rewrite, y = 50 + 20x.

If we compare it with standard linear equation, y = mx + c

where m --> slope of line

c --> y-intercept

then slope of equation (1) is equals to 'b' and b = 20. In this situation, the slope of line tell me that except the booking fee, the additional cost per person who will attend the party is equals to $20. Hence, the required answer is option(b).

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Complete question:

Emmett is planning a party at an escape room. The escape room will charge Emmett a booking fee and an additional cost per person attending the party.

This situation can be modeled as a linear relationship, see above graph. What does the slope of the line tell you about the situation?

A. The booking fee is $50

B. After the booking fee the party will cost $20 per person

C. After the booking fee the party will cost $25 per person

D. Including the bookin fee it would cost emmett $250 total for 10 people to attend.

B. After the booking fee the party will cost $20 per person

When the sum of the angles is exactly 360 degrees, there is no overlap or gap at the vertex.

What is a polygon ?

A polygon is a two-dimensional closed shape made up of line segments. It has three or more straight sides and angles. Some common examples of polygons are triangles, rectangles, squares, pentagons, hexagons, and octagons.

The properties of a polygon, such as the number of sides, angles, and vertices, depend on its type and shape.

The condition that will not create a gap or an overlap at a vertex of polygons is "The sum of the measures of the angles is equal to 360°."

When the sum of the angles is exactly 360 degrees, there is no overlap or gap at the vertex.

This condition is also known as the angle sum property of polygons, and it states that the sum of the interior angles of a polygon with n sides is equal to (n-2) times 180 degrees.

Therefore, When the sum of the angles is exactly 360 degrees, there is no overlap or gap at the vertex.

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The arrow travels a distance of 2145 meters into the bale of paper after 3 seconds.

The distance an arrow travels into a bale of paper is given by the equation \[ s(t)=2401-(7-t)^{4}, 0 \leq t \leq 7 \]. To find the distance the arrow travels at a specific time, we simply plug in the value of \( t \) into the equation and solve for \( s \).

For example, if we want to find the distance the arrow travels after 3 seconds, we would plug in \( t=3 \) into the equation:
\[ s(3)=2401-(7-3)^{4} \]
\[ s(3)=2401-(4)^{4} \]
\[ s(3)=2401-256 \]
\[ s(3)=2145 \]

Similarly, we can find the distance the arrow travels at any time \( t \) by plugging in the value of \( t \) into the equation and solving for \( s \). This will give us the distance the arrow travels horizontally into the bale of paper at that specific time.

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The expression are classified as follows

1. Decay

2. Growth

3. Growth

3.  Decay

4. Growth

5. Decay

6. Decay

Exponential function refers to functions of the form

f(x) = a(b)^x

where

the starting = a

the base = b

the exponents = x

The base is what determines exponential growth or decay

when the base is greater than 1 we have exponential growth

when the base is 0 < x < 1 we have exponential decay

comparing the base of the functions shows that

1. y = 500(0.30)  decay

2. y = 500(1.70) growth

3. y = 0.3(500) growth

3. y=500(0.30) - 6  decay

4.y = 0.3(1.7)¹-2 growth

5. y = 500(0.30)+2 decay

6. y = 500(0.30)-6  decay

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As a result, the park's estimated area, calculated using a trapezoidal sum with three equally spaced subintervals, is 50 square miles.

A quadrilateral with at least one set of parallel edges is known as a trapezoid. The bases and legs of a trapezoid are referred to by their parallel and non-parallel edges, respectively. The space between the bases of a trapezoid measured perpendicularly is its height. A = (1/2)h(b1 + b2), where A is the area, h is the height, and b1 and b2 are the lengths of the two bases, is the formula for a trapezoid's area.

given

By calculating the function's average at the left and right ends of the subinterval, it is possible to determine the height of each trapezoid.

A1 = (1/2) * (base1 + base2) * height

= (1/2) * (2 + 2) * 9 = 18 is the area of the first trapezoid.

A2 = (1/2) * (base1 + base2) * height

= (1/2) * (2 + 2) * 11 = 22 is the formula for the area of the second trapezoid.

The third trapezoid's area is given by

A3 = (1/2) * (base1 + base2) * (1/2) * (2 + 2). * 10

= 10

The three trapezoids' combined areas make up the entire area under the curve, which roughly corresponds to the size of the park:

A total = A1 + A2 + A3 

= 50

As a result, the park's estimated area, calculated using a trapezoidal sum with three equally spaced subintervals, is 50 square miles.

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Zach still has to save 1/12 of the total amount after this month.

To find out how much of the total amount Zach still has to save after this month, we need to calculate how much he has already saved and how much he will save from this month's paycheck.

First, we know that Zach has already saved 2/3 of the money he will need for his trip. So, if we let x be the total amount he needs for the trip, we can write this as:
Already saved = 2/3 * x

Next, we know that Zach plans to put aside 3/4 of what he has left to save from this month's paycheck. Since he has already saved 2/3 of the total amount, he has 1/3 of the total amount left to save. So, the amount he will save from this month's paycheck is:
Amount from paycheck = 3/4 * (1/3 * x) = 1/4 * x

Now, we can add these two amounts to find out how much Zach has saved in total:
Total saved = Already saved + Amount from paycheck
Total saved = (2/3 * x) + (1/4 * x) = (8/12 * x) + (3/12 * x) = 11/12 * x

Finally, we can subtract this amount from the total amount he needs to find out how much he still has to save:
Amount still to save = Total amount - Total saved
Amount still to save = x - (11/12 * x) = (12/12 * x) - (11/12 * x) = 1/12 * x

So, Zach still has to save 1/12 of the total amount after this month.  

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A trinomial with the leading coefficient -10 in terms of x in simplest form is -10x^2.

A trinomial with a leading coefficient of -10 in terms of x can be written as:
-10x^2 + bx + c

Where b and c are any real numbers. The simplest form of this trinomial would be:
-10x^2 + 0x + 0

Which can be simplified to:
-10x^2

A trinomial is an algebraic expression that has three non-zero terms and has more than one variable in the expression. This expression has three terms. Therefore, this expression is called a trinomial.

A trinomial is a type of polynomial but with three terms.

The examples of trinomials are: x + y + 7. ab + a +b,

3x+5y+8z with x, y, z variables

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Answer:

5/33

Step-by-step explanation:

5/6 divided by 11/2

Invert the divider

5/6 x 2/ 11

10/66

Reduce to smallest fraction

5/33

Answer:

5/33

Step-by-step explanation:

Apply the fraction rule: a/b ÷ c/d = (a × d) ÷ (b × c) for 5/6 ÷ 11/2

a = 5

b = 6

c = 11

d = 2

(5 × 2) ÷ (6 × 11)

For "(6 × 11)", break 6 down into "2 × 3", so that you can cancel out the common factor, because there is also a 2 in "5 × 2".

= (5 × 2) ÷ (2 × 3 × 11)         -- cancel out the 2's

= 5 ÷ (3 × 11)     *****3 × 11 = 33*****

= 5/33

Answer:

First, let's find the total number of crayons produced each day:

500 boxes/day x 16 crayons/box = 8,000 crayons/day

Since each crayon uses 400 milliliters of melted wax, the total amount of wax used in one day can be found by multiplying the number of crayons by the amount of wax used per crayon and converting from milliliters to liters:

8,000 crayons/day x 400 mL/crayon x 1 L/1000 mL = 3,200 liters/day

Therefore, Bright Designs Crayon Factory uses 3,200 liters of melted wax each day.

Step-by-step explanation:

Answer:

Bright Designs Crayon Factory uses 3,200 liters of melted wax each day.

Step-by-step explanation:

a. The standard error of the mean would be 9.49.

b. The probability that the sample mean will be less than $125 is 0.1539.

c. The probability that the sample mean will be more than $145 is 0.1401.

d. The probability that the sample mean will be between $120 and $160 is 0.9356.

e. The symmetrical interval that includes 95% of the sample means is ($116.11, $153.31).

a. The standard error of the mean is calculated using the formula:

SE = σ / √n

where σ is the standard deviation and n is the sample size.

In this case, σ = $30 and n = 10. So, the standard error of the mean is:

SE = 30 / √10

SE = 9.49

b. To find the probability that the sample mean will be less than $125, we need to calculate the z-score:

z = (x - μ) / SE

where x is the sample mean, μ is the population mean, and SE is the standard error of the mean.

In this case, x = $125, μ = $134.71, and SE = 9.49. So, the z-score is:

z = (125 - 134.71) / 9.49

z = -1.02

Using a z-table, we find that the probability of getting a z-score less than -1.02 is 0.1539. So, the probability that the sample mean will be less than $125 is 0.1539.

c. To find the probability that the sample mean will be more than $145, we need to calculate the z-score:

z = (x - μ) / SE

where x is the sample mean, μ is the population mean, and SE is the standard error of the mean.

In this case, x = $145, μ = $134.71, and SE = 9.49. So, the z-score is:

z = (145 - 134.71) / 9.49

z = 1.08

Using a z-table, we find that the probability of getting a z-score less than 1.08 is 0.8599. So, the probability that the sample mean will be more than $145 is 1 - 0.8599 = 0.1401.

d. To find the probability that the sample mean will be between $120 and $160, we need to calculate the z-scores for both values:

z1 = (x1 - μ) / SE

z2 = (x2 - μ) / SE

where x1 and x2 are the sample means, μ is the population mean, and SE is the standard error of the mean.

In this case, x1 = $120, x2 = $160, μ = $134.71, and SE = 9.49. So, the z-scores are:

z1 = (120 - 134.71) / 9.49

z1 = -1.55

z2 = (160 - 134.71) / 9.49

z2 = 2.67

Using a z-table, we find that the probability of getting a z-score less than -1.55 is 0.0606 and the probability of getting a z-score less than 2.67 is 0.9962. So, the probability that the sample mean will be between $120 and $160 is 0.9962 - 0.0606 = 0.9356.

e. To find the symmetrical interval that includes 95% of the sample means, we need to use the formula:

x = μ ± z*SE

where x is the sample mean, μ is the population mean, z is the z-score, and SE is the standard error of the mean.

In this case, μ = $134.71, z = 1.96 (for a 95% confidence interval), and SE = 9.49. So, the symmetrical interval is:

x = 134.71 ± 1.96*9.49

x = 134.71 ± 18.6

x = (116.11, 153.31)

So, the symmetrical interval that includes 95% of the sample means is ($116.11, $153.31).

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The solution of the exponential equation is x = 0.35

An exponential equation is an equation that contains exponents.

Since we have the exponential equation

4ˣ = 8ˣ - 1

We proceed to solve as follows

4ˣ = 8ˣ - 1

(2²)ˣ = (2⁴)ˣ - 1

(2ˣ)² = (2ˣ)⁴ - 1

Let 2ˣ = y

So, we have that

y² = y⁴ - 1

Re- arranging, we have that

y⁴ - y² - 1 = 0

Also, let y² = p. So, we have that

p² - p - 1 = 0

Now, we find p using the quadratic formula.

[tex]p = \frac{-b +/-\sqrt{b^{2} - 4ac} }{2a}[/tex]

where a = 1 b = -1 and c = -1

So, [tex]p = \frac{-(-1) +/-\sqrt{(-1)^{2} - 4(1)(-1)} }{2(1)}\\= \frac{1 +/-\sqrt{1 + 4} }{2}\\= \frac{1 +/-\sqrt{5} }{2}\\p = \frac{1 + 2.24 }{2} or p = \frac{1 - 2.24 }{2}\\p = \frac{3.24 }{2} or p = \frac{-1.24 }{2}\\p = 1.62 or p = -0.62[/tex]

We ignore the negative value.

So, p = 1.62

y² = 1.62

y = ±√1.62

y = ±1.273

Since y = 2ˣ, we have that

2ˣ = ±1.273

We ignore the negative value since the value of y cannot be negative.

So, 2ˣ = 1.273

Taking natural logarithm of both sides, we have that

㏑2ˣ = ㏑1.273

x㏑2 = ㏑1.273

x = ㏑1.273/㏑2

= 0.2412/0.693

= 0.348

≅ 0.35

So, x = 0.35

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The value of the variable x for the similar triangles is equal to 3

The triangles PTQ and STR are similar, this implies that the length ST of the smaller triangle is similar to the length PT of the larger triangle

and the length RT of the smaller triangle is similar to the length QT of the larger triangle

so;

5/(2x + 9) = 7/(3x + 12)

5(3x + 12) = 7(2x + 9) {cross multiplication}

15x + 60 = 14x + 63

15x - 14x = 63 - 60 {collect like terms}

x = 3.

Therefore, the value of the variable x for the similar triangles is equal to 3

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