GEOMETRY 100 POINTS

FIND X​

The measure of the angle m∠DAX on the straight line CX is equal to 140°

Angles that are on a straight line involves the sum of angles that can be arranged together so that they form a straight line. Angles on a straight line when added together sum up to 180°.

Given that AW bisects the angle m∠CAD and angle m∠CAW is equal to 20°, then;

m∠CAD = 2 × 20°

m∠CAD = 40°

m∠DAX + m∠CAD = 180° {sum of angles on a straight line}

m∠DAX + 40° = 180°

m∠DAX = 180° - 40° {subtract 40° from both sides}

m∠DAX = 140°

Therefore, the measure of the angle m∠DAX on the straight line CX is equal to 140°

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Answer: The measure of the angle m∠DAX on the straight line CX is equal to 140°

Step-by-step explanation: the gut below beat me to it

Answer:

y=10

Step-by-step explanation:

y=3x+4

substitute for x

y=3(2)+4

y=6+4

y=10

Answer:

x = 3/2 or x = -1

Step-by-step explanation:

2x² - x - 3 = 0

2*(-3) = -6

Factors of -6:

(-1, 6), (1, -6), (-2, 3), (2, -3)

We need to find a pair that adds up to the co-eff of x which is (-1)

Factors :(2,-3)

2 - 3 = -1

so, 2x² - x - 3 = 0 can be written as:

2x² + 2x - 3x - 3 = 0

⇒ 2x(x + 1) -3(x + 1) = 0

⇒ (2x - 3)(x + 1) = 0

⇒ 2x - 3 = 0 or

x + 1 = 0

⇒ 2x = 3 or x = -1

⇒ x = 3/2 or x = -1

Step-by-step explanation:

Since it's five (5) sided it is a pentagon.

First find the sum of the interior angle of a Pentagon .

S = ( n -2 ) × 180°

n represent the number of sides

S = ( 5 -2 ) × 180°

S = 3×180°

S = 540°

5x+2+ 10x-3 +7x-11 +8x-19 + 13x -31 = 540°

43x - 62 = 540°

43x = 540° + 62

43x = 602°

x = 14°

Answer : 14°

The calculated ratio of the flowers are

From the question, we have the following parameters that can be used in our computation:

Roses = 5

Flowers = 12

This means that

Daises = 12 - 5

Evaluate

Daises = 7

Next, we have the ratio to be

Daises : Roses = 7 : 5

Also, we have

Flower : Daises = 12 : 5

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Hello!

area

= 2*25 + (20 - 2)*(25-8)

= 50cm² + 306cm²

= 356cm²

The complete frequency table is

Interval          Tally     Frequency

0 - 1                 |||||           5

2 - 3                               0

4 - 5                 ||||           4

6 - 7                 ||||||||         8

8 - 9                 ||||||          6

From the question, we have the following parameters that can be used in our computation:

The set of data

The tally is used to calculate the frequencies of the readings using marks

So, we have

Interval          Tally     Frequency

0 - 1                 |||||           5

2 - 3                               0

4 - 5                 ||||           4

6 - 7                 ||||||||         8

8 - 9                 ||||||          6

The above represents the complete frequency table

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well, if we take a peek at the number line above, from one integer to the next there are 5 slots or divisions, meaning each division is 1/5.

what's the distance of DE?

well, D is at -1 plus 4 slots, that means -1 and we move it further by 4/5, that makes it -1⅘.

now E is at 2 plus 2 slots more, so that means 2 and we move it further by 2/5 or 2⅖.

to get the distance we simply get their difference, not much to it.

[tex]\stackrel{mixed}{1\frac{4}{5}}\implies \cfrac{1\cdot 5+4}{5}\implies \stackrel{improper}{\cfrac{9}{5}}~\hfill \stackrel{mixed}{2\frac{2}{5}} \implies \cfrac{2\cdot 5+2}{5} \implies \stackrel{improper}{\cfrac{12}{5}} \\\\[-0.35em] ~\dotfill\\\\ |DE|\implies \left| -\cfrac{9}{5}-\cfrac{12}{5} \right|\implies \left| -\cfrac{21}{5} \right|\implies \cfrac{21}{5}\implies \text{\LARGE 4.2}[/tex]

how about the midpoint of DE?

well, we can move from the left over to the midpoint or we can move from  the right, let's move from the  right, so hmmm let's take half of 21/5 and subtract it from E, that's where the midpoint is.

[tex]\stackrel{ E }{\cfrac{12}{5}}~~ - ~~\stackrel{ \textit{half of DE} }{\left( \cfrac{21}{5}\cdot \cfrac{1}{2} \right)}\implies \cfrac{12}{5}-\cfrac{21}{10}\implies \cfrac{(2)12~~ - ~~(1)21}{\underset{\textit{using this LCD}}{10}} \\\\\\ \cfrac{24-21}{10}\implies \cfrac{3}{10}\implies \text{\LARGE 0.3}[/tex]

Part (a)

Answer: Constant of proportionality = 5/8

Reason:

The general template equation is y = kx where k is the constant of proportionality. It is the slope of the line.

The direct proportion line must pass through the origin. In other words, the y intercept must be zero.

=====================================

Part (b)

Answer: Not Proportional

Reason:

The y intercept isn't zero.

Plug x = 0 into the equation to find y = 1 is the y intercept. This graph does not pass through the origin.

The probability of selecting a junior who takes the bus is P (Junior who takes the bus) = 12/200 = 0.06Hence, the probability that a randomly selected student is a junior who takes the bus is 0.06 or 6/100.

The given frequency table on how students get to school among the high school students is represented in the below table:Transportation Walk Bike BusDriveTotalGrade 9 11 10 14 15 50Grade 10 10 7 13 20 50Grade 11 8 6 12 24 50Grade 12 5 8 8 29 50 Total 34 31 47 88 200Given data from the above frequency table, we are interested in finding the probability of a randomly selected student being a junior who takes the bus.SolutionWe know that the total number of students is 200, and the total number of junior students is 50. Hence the probability of selecting a junior is P (Junior) = 50/200 = 0.25Similarly, the number of students who take the bus is 47 and the number of junior students who take the bus is 12.

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Answer:

---------------

Let the numbers be s and l.

We are given that:

Set up equations:

Eliminate l:

Solve for s:

Find l:

Answer:

Step-by-step explanation:

Answer:

[tex]\displaystyle \frac{75}{2}[/tex] or [tex]37.5[/tex]

Step-by-step explanation:

We can answer this problem geometrically:

[tex]\displaystyle \int^6_{-4}f(x)\,dx=\int^1_{-4}f(x)\,dx+\int^3_1f(x)\,dx+\int^6_3f(x)\,dx\\\\\int^6_{-4}f(x)\,dx=(5*5)+\frac{1}{2}(2*5)+\frac{1}{2}(3*5)\\\\\int^6_{-4}f(x)\,dx=25+5+7.5\\\\\int^6_{-4}f(x)\,dx=37.5=\frac{75}{2}[/tex]

Notice that we found the area of the rectangular region between -4 and 1, and then the two triangular areas from 1 to 3 and 3 to 6. We then found the sum of these areas to get the total area under the curve of f(x) from -4 to 6.

Answer:

[tex]\dfrac{75}{2}[/tex]

Step-by-step explanation:

The value of a definite integral represents the area between the x-axis and the graph of the function you’re integrating between two limits.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{De\:\!finite integration}\\\\$\displaystyle \int^b_a f(x)\:\:\text{d}x$\\\\\\where $a$ is the lower limit and $b$ is the upper limit.\\\end{minipage}}[/tex]

The given definite integral is:

[tex]\displaystyle \int^6_{-4} f(x)\; \;\text{d}x[/tex]

This means we need to find the area between the x-axis and the function between the limits x = -4 and x = 6.

Notice that the function touches the x-axis at x = 3.

Therefore, we can separate the integral into two areas and add them together:

[tex]\displaystyle \int^6_{-4} f(x)\; \;\text{d}x=\int^3_{-4} f(x)\; \;\text{d}x+\int^6_{3} f(x)\; \;\text{d}x[/tex]

The area between the x-axis and the function between the limits x = -4 and x = 3 is a trapezoid with bases of 5 and 7 units, and a height of 5 units.

The area between the x-axis and the function between the limits x = 3 and x = 6 is a triangle with base of 3 units and height of 5 units.

Using the formulas for the area of a trapezoid and the area of a triangle, the definite integral can be calculated as follows:

[tex]\begin{aligned}\displaystyle \int^6_{-4} f(x)\; \;\text{d}x & =\int^3_{-4} f(x)\; \;\text{d}x+\int^6_{3} f(x)\; \;\text{d}x\\\\& =\dfrac{1}{2}(5+7)(5)+\dfrac{1}{2}(3)(5)\\\\& =30+\dfrac{15}{2}\\\\& =\dfrac{75}{2}\end{aligned}[/tex]

Answer:

[tex]f'(\frac{1}{3\sqrt{e}})=\frac{1}{2}[/tex]

Step-by-step explanation:

Find f'(x) using Product Rule

[tex]f(x)=x\ln(3x)\\f'(x)=\ln(3x)+3x(\frac{1}{3x})\\f'(x)=\ln(3x)+1\\\\f'(\frac{1}{3\sqrt{e}})=\ln(3\cdot\frac{1}{3\sqrt{e}})+1\\\\f'(\frac{1}{3\sqrt{e}})=\ln(\frac{1}{\sqrt{e}})+1\\\\f'(\frac{1}{3\sqrt{e}})=\ln(e^{-\frac{1}{2}})+1\\\\f'(\frac{1}{3\sqrt{e}})=-\frac{1}{2}\ln(e)+1\\\\f'(\frac{1}{3\sqrt{e}})=-\frac{1}{2}+1\\\\f'(\frac{1}{3\sqrt{e}})=\frac{1}{2}[/tex]

Answer:

[tex]f'\left(\dfrac{1}{3\sqrt{e}}\right)=\dfrac{1}{2}[/tex]

Step-by-step explanation:

Given function:

[tex]f(x)=x\ln(3x)[/tex]

To find f'(x), differentiate the given function using the product rule.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

[tex]\textsf{Let\;$u=x^2}[/tex][tex]\textsf{Let\;$u=x$}\implies \dfrac{\text{d}u}{\text{d}x}=1[/tex]

[tex]\textsf{Let\;$v=\ln(3x)$}\implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{1}{3x}\cdot 3=\dfrac{1}{x}[/tex]

Input the values into the product rule to differentiate the function:

[tex]\begin{aligned}\dfrac{\text{d}y}{\text{d}x}&=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}\\\\&=x \cdot \dfrac{1}{x}+\ln(3x) \cdot 1\\\\&=1+\ln(3x)\end{aligned}[/tex]

To find the value of f'(1/(3√e)), substitute x = 1/(3√e) into the differentiated function:

[tex]\begin{aligned}f'\left(\dfrac{1}{3\sqrt{e}}\right)&=1+\ln\left(3\left(\dfrac{1}{3\sqrt{e}}\right)\right)\\\\&=1+\ln\left(\dfrac{1}{\sqrt{e}}\right)\\\\&=1+\ln e^{-\frac{1}{2}}\\\\&=1-\dfrac{1}{2}\ln e\\\\&=1-\dfrac{1}{2}(1)\\\\&=1-\dfrac{1}{2}\\\\&=\dfrac{1}{2}\end{aligned}[/tex]

[tex]\hrulefill[/tex]

Differentiation rules used:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Differentiating $ax$}\\\\If $y=ax$, then $\dfrac{\text{d}y}{\text{d}x}=a$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{6 cm}\underline{Differentiating $\ln(f(x))$}\\\\If $y=\ln(f(x))$, then $\dfrac{\text{d}y}{\text{d}x}=\dfrac{1}{f(x)}\cdot f'(x)$\\\end{minipage}}[/tex]

Answer:

x- intercept = - 8 , y- intercept = 4

Step-by-step explanation:

to find the x- intercept let y = 0 in the equation and solve for x

x - 2(0) = - 8

x - 0 = - 8

x = - 8 ← y- intercept

to find the y- intercept let x = 0 in the equation and solve for y

0 - 2y = - 8

- 2y = - 8 ( divide both sides by - 2 )

y = 4 ← y- intercept

The population variance of the data set is 16, and the population standard deviation is 4. These measures give an indication of how spread out the numbers are from the mean.

To find the population variance and standard deviation of a set of numbers, you can follow these steps:

Step 1: Find the mean (average) of the data set. In this case, the data set is 8, 11, 15, 17, and 19. The mean is calculated by summing up all the numbers and dividing by the total count. In this case, the mean is (8 + 11 + 15 + 17 + 19) / 5 = 14.

Step 2: Subtract the mean from each number and square the result. For example, subtracting 14 from 8 gives (-6)^2 = 36.

Step 3: Repeat Step 2 for each number in the data set. The squared differences for the given data set are 36, 9, 1, 9, and 25.

Step 4: Find the sum of all the squared differences. In this case, the sum is 36 + 9 + 1 + 9 + 25 = 80.

Step 5: Divide the sum of squared differences by the total count of numbers to calculate the population variance. The population variance is 80 / 5 = 16.

Step 6: Take the square root of the population variance to find the population standard deviation. The population standard deviation is √16 = 4.

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Answer:

arc PN = 136°

Step-by-step explanation:

the inscribed angle PLN is half the measure of its intercepted arc PN

then PN is twice the inscribed angle PLN , that is

PN = 2 × 68° = 136°

Answer:

1) x = 43.33

2) x =  tan⁻¹(4/3)  = 53.13

Step-by-step explanation:

1) Formula: cos(A) = sin(90 - A)

sin(2x - 30) = cos(x - 10)

⇒ sin(2x - 30) = sin(90 - (x - 10))

⇒ sin(2x - 30) = sin(90 - x + 10)

⇒ sin(2x - 30) = sin(100 - x)

⇒ 2x - 30 = 100 - x

⇒ 3x = 130

⇒ x = 130/3

x = 43.33

2) 4sinx . cosx - 3sin²x = 0

⇒ 4sinx . cosx = 3sin²x

⇒ 4cosx = 3sinx

⇒ [tex]\frac{4}{3} =\frac{sinx}{cosx}[/tex]

⇒ [tex]tan x = \frac{4}{3}[/tex]

⇒ x =  tan⁻¹(4/3)

⇒ x = 53.13

Answer:

[tex]\textsf{1)} \quad x \approx 43.33^{\circ}[/tex]

[tex]\textsf{2)} \quad \boxed{\begin{aligned}x &= \pi n \;\text{radians}\\x&=0.93+\pi n\; \text{radians}\end{aligned}}\quad \boxed{\begin{aligned}x &= 180^{\circ}n\\x&=53.13^{\circ}+180^{\circ} n\; \end{aligned}}[/tex]

Step-by-step explanation:

Given trigonometric equation:

[tex]\sin(2x - 30^{\circ}) = \cos(x - 10^{\circ})[/tex]

To solve the given trigonometric equation, we can use the following trigonometric identity:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Trigonometric identity} \\\\$\cos (\theta)=\sin(90^{\circ}-\theta)$\\\end{minipage}}[/tex]

Apply the trigonometric identity to the right side of the equation:

[tex]\begin{aligned}\sin(2x - 30^{\circ}) &= \cos(x - 10^{\circ})\\\\&= \sin(90^{\circ}-(x - 10^{\circ}))\\\\&= \sin(90^{\circ}-x +10^{\circ})\\\\&= \sin(100^{\circ}-x)\end{aligned}[/tex]

Since the sine function is equal, we can equate the angles:

[tex]2x - 30^{\circ}=100^{\circ}-x[/tex]

Now simplify and solve for x:

[tex]\begin{aligned}2x - 30^{\circ}&=100^{\circ}-x\\\\2x - 30^{\circ}+x&=100^{\circ}-x+x\\\\3x - 30^{\circ}&=100^{\circ}\\\\3x - 30^{\circ}+30^{\circ}&=100^{\circ}+30^{\circ}\\\\3x&=130^{\circ}\\\\\dfrac{3x}{3}&=\dfrac{130^{\circ}}{3}\\\\x&=\left(\dfrac{130}{3}\right)^{\circ}\\\\x&\approx 43.3^{\circ}\; \sf (nearest\;tenth)\end{aligned}[/tex]

Therefore, the solution to the equation sin(2x - 30°) = cos(x - 10°) is approximately x = 43.33°.

[tex]\hrulefill[/tex]

Given trigonometric equation:

[tex]4 \sin x \cos x-3\sin^2x=0[/tex]

Factor out the common term sin(x):

[tex]\sin x(4 \cos x-3\sin x)=0[/tex]

According to the zero product property, one of the factors must be equal to zero for the equation to hold.

Set each factor equal to zero and solve for x.

Factor 1

[tex]\sin x=0[/tex]

According to the unit circle, sin(x) = 0 when x = 0 and x = π.

As the sine function is periodic with a period of 2π, the solutions to sin(x) = 0 are:

[tex]x=0+2\pi n, \;\;x=\pi + 2\pi n[/tex]

Therefore, x is any multiple of π, where n is an integer:

[tex]\boxed{x = \pi n}[/tex]

Factor 2

[tex]\begin{aligned}4\cos x - 3 \sin x & = 0\\\\4 \cos x & = 3 \sin x\\\\\dfrac{4}{3}&=\dfrac{\sin x}{\cos x}\\\\\dfrac{4}{3}&=\tan x\\\\\implies x&=\arctan\left(\dfrac{4}{3}\right)\\\\x&=0.92729...\end{aligned}[/tex]

As the tangent function is periodic with a period of π, the solutions are:

[tex]\boxed{x=0.92729...+\pi n}[/tex]

where n is an integer.

Therefore, the solutions to the equation 4sin(x)cos(x) - 3sin²(x) = 0 are:

[tex]\boxed{\begin{aligned}x &= \pi n \;\text{radians}\\x&=0.93+\pi n\; \text{radians}\end{aligned}}[/tex]    [tex]\boxed{\begin{aligned}x &= 180^{\circ}n\\x&=53.13^{\circ}+180^{\circ} n\; \end{aligned}}[/tex]

(where n is an integer)

Answer:

The selling price of the diamond necklace at Shamin Jewelers after 10% discount is:

$442 * 0.9 = $397.80

The selling price of the same necklace at the competitor's store after 34% and 14% discount is:

$527 * 0.66 * 0.86 = $247.08

So, Shamin Jewelers needs to offer an additional discount to meet the competitor's price:

$397.80 - $247.08 = $150.72

To calculate the additional rate of discount, we divide the difference by the original selling price at Shamin Jewelers and multiply by 100:

($150.72 / $442) * 100 = 34.11%

Therefore, Shamin Jewelers must offer an additional 34.11% discount to meet the competitor's price.

Step-by-step explanation:

Answer:

p which is the y axis is -1

Answer:

x = 6

∠D = 127

Step-by-step explanation:

In a parallelogram, opposite angles are equal and adjacent angles add to 180

⇒ ∠A = ∠C and ∠C + ∠D = 180

∠A = ∠C

⇒ 13x - 25 = 9x - 1

⇒ 13x - 9x = 25 - 1

⇒ 4x = 24

⇒ x = 24/4

⇒ x = 6

∠C =  9x - 1

= 9(6) - 1

= 54 - 1

= 53

∠C + ∠D = 180

⇒ ∠D = 180 - ∠C

= 180 - 53

= 127

The expansion of the function [tex](3x - 4)^4[/tex] simplifies to [tex]81x^4 - 432x^3 + 864x^2 - 768x + 256.[/tex]

To expand the function [tex]f(x) = (3x - 4)^4[/tex], we can use the binomial theorem. According to the binomial theorem, for any real numbers a and b and a positive integer n, the expansion of [tex](a + b)^n[/tex] can be written as:

[tex](a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^{(n-1)} b^1 + C(n, 2)a^{(n-2)} b^2 + ... + C(n, n-1)a^1 b^{(n-1)} + C(n, n)a^0 b^n[/tex]

where C(n, k) represents the binomial coefficient, which is given by C(n, k) = n! / (k!(n-k)!).

Applying this formula to our function [tex]f(x) = (3x - 4)^4[/tex], we have:

[tex]f(x) = C(4, 0)(3x)^4 (-4)^0 + C(4, 1)(3x)^3 (-4)^1 + C(4, 2)(3x)^2 (-4)^2 + C(4, 3)(3x)^1 (-4)^3 + C(4, 4)(3x)^0 (-4)^4[/tex]

Simplifying each term, we get:

[tex]f(x) = 81x^4 + (-432x^3) + 864x^2 + (-768x) + 256[/tex]

Therefore, the expanded form of the function [tex]f(x) = (3x - 4)^4[/tex] is [tex]81x^4 - 432x^3 + 864x^2 - 768x + 256[/tex].

Note that the coefficient of [tex]x^3[/tex] is -432, the coefficient of [tex]x^2[/tex] is 864, the coefficient of x is -768, and the constant term is 256.

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Note the complete question is

i) 4 more gallons of petrol will be needed to complete the journey.

ii) The cost of the petrol for the 420 km journey is GH¢55.00.

i) To determine the number of gallons of petrol needed to complete the journey, we can calculate the total distance that can be covered with the available petrol and then subtract it from the total distance of the journey.

Given that the car consumes 1 gallon of petrol for every 30 km, we can calculate the distance that can be covered with 10 gallons of petrol by multiplying 10 (gallons) by 30 (km/gallon):

Distance covered with 10 gallons = 10 * 30 = 300 km

To find the remaining distance that needs to be covered, we subtract the distance covered with the available petrol from the total distance of the journey:

Remaining distance = Total distance - Distance covered with available petrol

Remaining distance = 420 km - 300 km = 120 km

Since the car consumes 1 gallon of petrol for every 30 km, we can determine the additional gallons of petrol needed by dividing the remaining distance by 30:

Additional gallons needed = Remaining distance / 30 = 120 km / 30 km/gallon = 4 gallons

Therefore, the driver will need 4 more gallons of petrol to complete the journey.

ii) To calculate the cost of the petrol for the journey of 420 km, we need to multiply the total number of gallons used for the journey by the cost per gallon.

Given that a gallon of petrol costs GH¢5.50, and the total number of gallons used for the journey is 10 (given in the problem), we can calculate the cost using the formula:

Cost of petrol = Total gallons used * Cost per gallon

Cost of petrol = 10 gallons * GH¢5.50/gallon = GH¢55.00

Therefore, the cost of the petrol for the journey of 420 km is GH¢55.00.

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Answer:

x = 10

Step-by-step explanation:

The diagonals of a rectangle are equal in length and bisect each other

⇒ JL = KM and

JN = LN = KN =MN

ΔJMN is an isosceles triangle since JN = MN

This means that the angles opposite to equal sides are equal

⇒ ∠JMN = ∠MJN

⇒ 7x - 2 = 3x + 38

⇒ 7x - 3x = 38 + 2

⇒ 4x = 40

⇒ x = 10

Answer:

x = 16

Step-by-step explanation:

Opposite sides are equal in a parallelogram

AD = BC

5x - 12 = 3x + 20

5x - 3x = 20 + 12

2x = 32

x = 32/2

x = 16

A) The slope between the points (0, -4) and (2, -1) is 1.5.

b)  the slope between the points (0, -4) and (4, -1) is 0.75.

c) The slope of 0.75 indicates that for every 1 unit increase in x, there is a 0.75 unit increase in y.

Part A: To calculate the slope between two points, we use the formula:

slope = (change in y) / (change in x)

Let's choose the points (0, -4) and (2, -1).

Change in y = -1 - (-4) = 3

Change in x = 2 - 0 = 2

slope = 3 / 2 = 1.5

Therefore, the slope between the points (0, -4) and (2, -1) is 1.5.

Part B: Let's choose different points from the table, such as (0, -4) and (4, -1).

Change in y = -1 - (-4) = 3

Change in x = 4 - 0 = 4

slope = 3 / 4 = 0.75

Thus, the slope between the points (0, -4) and (4, -1) is 0.75.

Part C: The slopes from parts A and B provide information about the relationship between the points.

In part A, where the slope was calculated as 1.5, we can see that as the x-values increase by 2 units, the y-values increase by 3 units. This suggests a positive relationship between the points, meaning that as x increases, y also increases. The slope of 1.5 indicates that for every 1 unit increase in x, there is a 1.5 unit increase in y.

In part B, where the slope was calculated as 0.75, we observe a similar positive relationship between the points. As the x-values increase by 4 units, the y-values increase by 3 units. The slope of 0.75 indicates that for every 1 unit increase in x, there is a 0.75 unit increase in y.

Overall, the positive slopes in both parts A and B suggest that the points on the table exhibit a positive linear relationship. This means that as x increases, y also increases, and the rate of change is consistent based on the slope.

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Sophia will drive a distance of 300 miles.

To find out how far Sophia will drive, we need to calculate 3/5 of the total trip distance of 500 miles.

1: Calculate the distance driven by Sophia

To find the distance driven by Sophia, we need to multiply the total trip distance by the fraction representing the portion she will drive. Sophia will drive the first 3/5 of the trip.

Distance driven by Sophia = (3/5) * 500 miles

2: Simplify the fraction

To calculate the distance, we simplify the fraction 3/5.

Distance driven by Sophia = (3/5) * 500 miles

                       = (3 * 500) / 5 miles

                       = 1500 / 5 miles

                       = 300 miles

Therefore, Sophia will drive a distance of 300 miles.

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The sets of side measurements that can form a triangle are: c. 8, 3, 11

d. 5, 11, 8

To determine which set of side measurements could be used to form a triangle, we need to apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's evaluate each set of side measurements:

a. 12, 23, 7

In this case, the sum of the lengths of the two shorter sides (12 and 7) is 19, which is less than the length of the longest side (23). Therefore, this set of side measurements cannot form a triangle.

b. 10, 7, 2

Here, the sum of the two shorter sides (10 and 2) is 12, which is greater than the length of the longest side (7). This set of side measurements also fails to satisfy the triangle inequality theorem and cannot form a triangle.

c. 8, 3, 11

The sum of the two shorter sides (8 and 3) is 11, which is greater than the length of the longest side (11). Therefore, this set of side measurements can form a triangle.

d. 5, 11, 8

In this case, the sum of the lengths of the two shorter sides (5 and 8) is 13, which is greater than the length of the longest side (11). Hence, this set of side measurements can also form a triangle.

In summary, the sets of side measurements that can form a triangle are:

c. 8, 3, 11

d. 5, 11, 8

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Step-by-step explanation: