help please!!!!!!!!!!

Answer:

(A) AA similarity

Step-by-step explanation:

In ΔABC,

∠A + ∠B + ∠C = 180

∠A + 27 + 90 = 180

∠A = 180 - 90 - 27

∠A = 63

Comparing ΔABC and ΔMNP,

∠A = ∠M = 63

∠C = ∠P = 90

Therfore, by AA property, the two triangles are similar

The measure of line segment x in the game board is approximately 13.6.

The secant-tangent power theorem states that "if a tangent and a secant are drawn from a common external point to a circle, then the product of the length of the secant segment and its external part is equal to the square of the length of the tangent segment.

It is expressed as:

( tangent segment )² = External part of the secant segment × Secant segment.

From the given figure:

Let;

Tangent segment = 12

Secant segment = 7 + x

External part of the secant segment = 7

Plug these values into the above formula and solve for x.

( tangent segment )² = External part of the secant segment × Secant segment.

12² = 7( 7 + x)

144 = 49 + 7x

7x = 144 - 49

7x = 95

x = 95/7

x = 13.6

Therefore, the value of x is approximately 13.6.

Option C) 13.6 is the correct answer.

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The arrow is at a height of 48 ft after approx. 3 - √6 s and after 3 + √6 s.

To find the time it takes for the arrow to reach a height of 48 ft, we can use the formula for the height of the arrow:

s = v0t - 16t^2

Here, s represents the height of the arrow, v0 is the initial velocity, and t is the time.

Given that the initial velocity, v0, is 96 ft/s and the height, s, is 48 ft, we can set up the equation:

48 = 96t - 16t^2

Now, let's solve this equation to find the time it takes for the arrow to reach a height of 48 ft.

Rearranging the equation:

16t^2 - 96t + 48 = 0

Dividing the equation by 16 to simplify:

t^2 - 6*t + 3 = 0

We now have a quadratic equation in the form of at^2 + bt + c = 0, where a = 1, b = -6, and c = 3.

Using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values:

t = (6 ± √((-6)^2 - 413)) / (2*1)

t = (6 ± √(36 - 12)) / 2

t = (6 ± √24) / 2

Simplifying the square root:

t = (6 ± 2√6) / 2

t = 3 ± √6

Therefore, the arrow reaches a height of 48 ft after approximately 3 + √6 seconds and 3 - √6 seconds.

In summary, the arrow takes approximately 3 + √6 seconds and 3 - √6 seconds to reach a height of 48 ft, assuming an initial velocity of 96 ft/s.

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Note the complete question is

The height of an arrow shot upward can be given by the formula s = v0*t - 16*t², where v0 is the initial velocity and t is time.How long does it take for the arrow to reach a height of 48 ft if it has an initial velocity of 96 ft/s?  

Solve (t-3)^2=6

The arrow is at a height of 48 ft after approx. ___ s and after ___ s.

Answer:

The product of each pair of complex conjugates is:

(3 + 8i)(3 – 8i) = (9 - 8^2) + i(9 + 8^2) = (-81) + i(99) = (-81) + i(11) = (-81) + 11i

And

(4 + 5i)(4 – 5i) = (16 - 25) + i(16 + 25) = (-9) + i(41) = (-9) + 41i

So, the products are:

(-81) + 11i

and

(-9) + 41i

Answer:

73

41

Step-by-step explanation:

a) The experimental probability of landing on red or yellow is:

P(red or yellow) = 33/40

b) The theoretical probability can be computed as follows:

P(red or yellow)  = 8/10

c) As the number of spins increases: Option A: we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

Theoretical probability represents the likelihood of an event occurring. The theoretical probability of getting heads is 1/2, because we know that the probability of heads and tails on a coin is equal. Experimental probability describes how often an event actually occurred in an experiment.  

a) The experimental probability of landing on red or yellow is:

P(red or yellow) = (20/40) + (13/40) = 33/40

b) The theoretical probability can be computed as follows:

P(red or yellow) = (4/10) + (4/10) = 8/10

c) As the number of spins increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal.

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To create a box from a flat piece of metal, you typically cut equal-sized squares from each corner, then fold up the sides. This effectively reduces the size of the base of the final box by twice the length of the side of the squares.

The original size of the piece of tin is 40 cm x 70 cm. Let's represent the length of the side of the squares cut from the corners as x (in cm). After the squares are removed, the size of the base of the box will be (40 - 2x) cm by (70 - 2x) cm.

The problem states that the base area of the box is 1984 cm^2. Thus, we can set up the following equation:

(40 - 2x) * (70 - 2x) = 1984.

Expanding this equation gives:

2800 - 140x - 80x + 4x^2 = 1984,

4x^2 - 220x + 2800 = 1984,

4x^2 - 220x + 816 = 0.

Dividing through by 4 gives:

x^2 - 55x + 204 = 0.

This quadratic equation can be solved using the quadratic formula, x = [ -b ± sqrt(b^2 - 4ac) ] / 2a. Here, a = 1, b = -55, and c = 204.

The discriminant, b^2 - 4ac = (-55)^2 - 4*1*204 = 3025 - 816 = 2209.

The square root of 2209 is 47.

Therefore, the solutions for x are:

x = [55 ± 47] / 2

x = 51, 4.

Since the length of the sides of the squares (x) must be less than half the length of the shorter side of the original tin piece (which is 40 cm / 2 = 20 cm), the valid solution is x = 4 cm. Thus, each square cut from the corners is 4 cm on a side.

We can make the decision as per rational or irrational from the solution of the equations.

A. The solution is irrational

B. The solution is rational

C. The solution is rational

D. The solution is irrational

A rational solution refers to a solution that can be expressed as a ratio of two integers, where the denominator is not zero. It is a value that satisfies an equation or inequality and can be written in the form of a fraction.

For the equation[tex]5x^2 + 4x - 14 = 0[/tex], we obtain the solution as;

-2/5 ±1/5√74

For the equation we have that[tex]4x^2 + 8x - 5 =0[/tex]. The solution would be x = 1/2 or -5/2

For the third equation we have that [tex]x^2 + x - 42 = 0[/tex] The solution is x = 6 and -7

For the fourth equation; is 3x^2 + 4x - 2= 0. The solution of the equation is -2/3 ± 1/3√10

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The members of the given set in this problem are given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

The union and intersection sets of multiple sets are defined as follows:

The intersection of the sets Y and Z for this problem is given as follows:

Y ∩ Z = {0, 6, 23, 26}

(which are the elements that belong to both of the sets).

The union of the above set with the set X is given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

(elements that belong to at least one of the sets).

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Answer: x=15

Step-by-step explanation:

If JH is a midsegment, then J is the midpoint of LK and H is the midpoint of KM. This also means that triangles JKH and triangles LKM are similar. Since H is the midpoint of KM, assume KM = 2y (KH=y). 30/x=2/1. Thus x=15.

Answer:

x = 15

Step-by-step explanation:

The midsegment of a triangle is a line segment that connects the midpoints of two sides of a triangle.

Therefore, if JH is the midsegment of ΔKLM then JH is parallel to LM, and triangles KJH and KLM are similar: ΔKJH ~ ΔKLM.

In similar triangles, corresponding sides are always in the same ratio.

Since JH is the midsegment of ΔKLM, then KJ is half the length of KL, and KH is half the length of KM. This means that JH is half the length of LM.

Given the measure of LM is 30:

[tex]x=\overline{JH}=\dfrac{30}{2}=15[/tex]

Answer: y = (-28)x + 14

Step-by-step explanation:

The equation for the line tangent at point (a, b) can be found using the formula y = mx + c, where m is the slope of the line and c is the y-intercept. At point (6, 14), we need to find the value of m.

We start by finding the derivative of the original function g(x). Its derivative is given by:

dg(x)/dx=7/(x^2-3x+1)

Then, substitute x = 6 into the derivative expression to obtain:

dg(6)/dx = d/dx [7 * ln|x-3| ] evaluated at x = 6 = 7/3

Next, evaluate the original function g(x) at x = 6 to get g(6) = 7 * ln |6 - 3| / (6 - 3) = 7 * ln 3.

Since we know the coordinates of the point of tangency (6, 14), we can substitute them into the general form of the linear equation y = mx + c:

14 = 7 * 6 + c

14 = 42 + c

c = -28

The final equation of the line tangent at point (6, 14) is therefore:

y = (-28)x + 14

(a)The experimental probability of rolling an odd number is 3/5.

(b) The theoretical probability of rolling an odd number is 1/2.

(c)The true statement is:

With a small number of rolls, it is not surprising when the experimental probability is much greater than the theoretical probability.

Probability is the likelihood of a desired event happening.

Experimental probability is a probability that relies mainly on a series of experiments.

Theoretical probability is the theory behind probability. To find the probability of an event, an experiment is not required. Instead, we should know about the situation to find the probability of an event occurring.

We have:

Kala rolled a number cube 20 times.

(a) From Kala's results, odd number (1, 3 and 5) appearance is:

4 + 4 + 4 = 12

Thus, the experimental probability of rolling an odd number will be:

12/20 = 3/5

(b) If the cube is fair, we have the numbers have equal chance of appearance.

Theoretically, an odd numbers (1, 3 and 5) will have the probability of 3 out of 6 as illustrated with the numbers below:

1, 2, 3, 4, 5, 6

The theoretical probability of rolling an odd number will be:

3/6 = 1/2

(c)The true statement is:

With a small number of rolls, it is not surprising when the experimental probability is much greater than the theoretical probability.

Because when conducting a small number of rolls with a fair cube, the experimental probability may not necessarily align closely with the theoretical probability.

This difference is expected due to the limited sample size. As the number of rolls increases, the experimental probability tends to converge towards the theoretical probability.

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The amount that can be withdrawn immediately after the last deposit is approximately $8,876.80.

We can use the formula for the future value of an ordinary annuity. The formula is given by:

[tex]FV = P * [(1 + r/n)^(^n^t^) - 1] / (r/n)[/tex]

Where

Given:

Plugging in the values, we have:

[tex]FV = 480 * [(1 + 0.045/2)^(^2*^8^) - 1] / (0.045/2)[/tex]

Calculating the expression inside the brackets

[tex](1 + 0.045/2)^(^2^*^8^) = 1.0225^1^6[/tex]≈ 1.4197

Substituting this value back into the formula:

FV = 480 * (1.4197 - 1) / (0.045/2)

FV = 480 * 0.4197 / 0.0225

FV ≈ $8,876.80

So, the amount that can be withdrawn immediately after the last deposit is approximately $8,876.80.

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The value of dy/dt  at x= 1 if y = [tex]3x^2[/tex] - 3 and dx/dt = 3 is 18.

To find dy/dt at x = 1, we need to differentiate the given equation y = [tex]3x^2[/tex] - 3 with respect to t and then substitute x = 1 and dx/dt = 3.

Let's begin by differentiating y = [tex]3x^2[/tex]- 3 with respect to t using the chain rule. Since y is a function of x and x is a function of t, we have:

dy/dt = (dy/dx) * (dx/dt)

Given dx/dt = 3, we can rewrite the equation as:

dy/dt = [tex](dy/dx) * 3[/tex]

Now, let's find the derivative of y = [tex]3x^2 - 3[/tex] with respect to x:

dy/dx = d/dx[tex](3x^2 - 3)[/tex]

= 6x

Substituting this back into the equation for dy/dt, we get:

dy/dt = [tex](6x) * 3[/tex]

= 18x

Finally, we can evaluate dy/dt at x = 1:

dy/dt at x = 1 = 18(1)

= 18

Therefore, the value of dy/dt at x = 1 is 18.

It's important to note that the value of dx/dt = 3 was given, and we used it to find dy/dt using the chain rule and the derivative of the equation with respect to x.


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Answer:

Diameter=20 cm, Area=100[tex]\pi[/tex]

Step-by-step explanation:

To find the diamter of a circle (given the radius), you multiply the radius by 2.  10cm*2=20 cm.  The area of a circle is the radius squared times pi.  10^2 times pi is equal to 100[tex]\pi[/tex].

Answer:

The Answer Will Be 16

Step-by-step explanation:

Beacuse , To Find Number of subsets we can use 2n Where 2 is number can't be changed but n is cardinal number so it's 16

type the whole number 16

(a) To compute the experimental probability of rolling a 1 or 6 from Latoya's results, we need to determine the number of times a 1 or 6 was rolled and divide it by the total number of rolls Latoya made.

Let's assume that Latoya rolled the dice 100 times and obtained 20 1's and 10 6's. The total number of successful outcomes (rolling a 1 or 6) is 20 + 10 = 30.

Therefore, the experimental probability of rolling a 1 or 6 is 30/100 = 0.3 or 30%.

(b) Assuming the cube is fair, the theoretical probability of rolling a 1 or 6 can be determined by considering the favorable outcomes (rolling a 1 or 6) divided by the total possible outcomes (rolling any number from 1 to 6).

Since there are two favorable outcomes (1 and 6) out of six possible outcomes (1, 2, 3, 4, 5, 6), the theoretical probability is 2/6 = 1/3 or approximately 0.3333.

(c) The correct statement is B. As the number of rolls increases, we expect the experimental and theoretical probabilities to become closer, though they might not be equal. This is due to the law of large numbers, which states that as more trials are conducted, the experimental probability tends to converge towards the theoretical probability. However, it is not necessary for them to be exactly equal. Random variations can cause some discrepancy, but with a larger number of rolls, the experimental probability should approach the theoretical probability more closely.

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The number of subsets in the given set is as follows:

128.

Considering a set with n elements, the number of subsets in the set is the nth power of 2, that is:

[tex]2^n[/tex]

The set in this problem is composed by integers between 2 and 8, hence it has these following elements:

{2, 3, 4, 5, 6, 7, 8}.

The set has four elements, meaning that n = 7, hence the number of subsets is given as follows:

[tex]2^7 = 128[/tex]

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Answer:

The 3.907 is approximate.

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

Explanation:

x = number of hours that elapse

y = f(x) = number of tokens

If we use a graphing tool like a TI84 or GeoGebra, then the approximate solution to -3(2)^(x+1) + 90 = 0 is roughly x = 3.907

At around 3.907 hours is when the number of tokens is y = 0. Therefore, this is the approximate upper limit for the domain. The lower limit is x = 0.

The domain spans from x = 0 to roughly x = 3.907, and we shorten that down to 0 ≤ x ≤ 3.907

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

Plug in x = 0 to find y = 84. This is the largest value in the range.

The smallest value is y = 0.

The range spans from y = 0 to y = 84, so we get 0 ≤ y ≤ 84

Keep in mind y is the number of tokens. A fractional amount of tokens does not make sense, so we must have y be a whole number 1,2,3,...,83,84.

The x value can be fractional because 3.907 hours for instance is valid.

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

Extra info:

Answer:

Step-by-step explanation:

                                                   error

Answer:

x ≈ 75.2

Step-by-step explanation:

using the tangent ratio in the right triangle

tan62° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{x}{40}[/tex] ( multiply both sides by 40 )

40 × tan62° = x , then

x ≈ 75.2 ( to the nearest tenth )

Answer:

x = - 1

Step-by-step explanation:

P = [tex]\frac{x-2}{x+1}[/tex]

the denominator of the rational function cannot be zero as this would make it undefined. Equating the denominator to zero and solving gives the value that x cannot be.

x + 1 = 0 ( subtract 1 from both sides )

x = - 1

P is undefined when x = - 1

x = 8.33 and the solution to the equation 12x = 100 is x = 8.33 (rounded to the nearest hundredth).

To solve the equation 12x = 100, we need to isolate the variable x. We can do this by dividing both sides of the equation by 12.

12x = 100

Dividing both sides by 12:

(12x)/12 = 100/12

Simplifying:

x = 100/12

Now, let's calculate the value of x.

x = 8.33 (rounded to the nearest hundredth)

Therefore, the solution to the equation 12x = 100 is x = 8.33 (rounded to the nearest hundredth).

Here's the step-by-step solution:

12x = 100 (given equation)

Divide both sides by 12: (12x)/12 = 100/12

Simplify: x = 8.33

Please note that the answer is rounded to the nearest hundredth as requested.

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If the bottle is completely full at 11:40 a.m., the statements that help explain this result include:

Exponential growth involves a pattern or process that increases the initial quantity or value over time at a constant rate.

In an exponential growth situation, the rate or speed of growth remains proportional to the initial population.

The reasons for choosing Options A, B, and D follow:

Option A: at 11:35 a.m., the bottle was 1/2 full.  If at 11:40 a.m. it is completely full, it implies that it takes 5 minutes to become full (11:40 a.m. - 11:35 a.m.)

Option B: At 11:30 a.m., the bottle was 1/4 full and 10 minutes later, 11:40 a.m. it was full, meaning that it doubled its volume after 10 minutes.

Option D: Exponential growth involves a constant rate of growth and can be expressed as a(1+r)ˣ, where a is the initial value or quantity, r is the growth rate, 1+r is the growth factor,, and x is the exponent in time.

Thus, in this situation, the correct options are A, B, and D.

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A) The bottle was 1/2 full at 11:35 a.m. and doubled again after 5 minutes means that at 11:40 a.m. the bottle is completely full.

B) The bottle was 1/4 full at 11:30 a.m. and doubled twice after 10 minutes means that at 11:40 a.m. the bottle is completely full.

C) The bottle was 1/2 full at 11:35 a.m. and more bacteria were added to fill the bottle means that the bottle is completely full but with bacteria present in it.

D) Exponential growth involves a constant multiplicative rate of change means that bottle level rises with a constant multiplicative rate of change.

E) Exponential growth involves a constant additive rate of change means that bottle level rises with a constant additive rate of change.

The points will  meet x when it is folded are C or E

A point is a location of an item by taking reference of the origin, in typical case we regard it at (0,0), At the origin value of both the coordinate position is considered to be Zero, In simple words point indicates how much it is above or below the coordinate axes.

The vertical distance from the x axis is referred to as the y coordinate, while the horizontal distance from the x axis is referred to as the x coordinate.

A green colored Net,

After folding x point will  meet the points = ?

When we start folding,

First step,

Side ABCD and DEFG will be folded,

C and E will meet with each other

Second Step,

Side ABCD and AGHI will be folded,

B and I will meet with each other

Third step,

Side DEFG and AGHI will be folded,

F and H will meet each other

Fourth step,

Upper portion AHI will cover the upper portion BCF of the box

X will at position of C or E

Hence, the X will meet

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The following question may be like this:

When this net is folded up, which two points meet X?

The quadratic function with the solutions given in the problem is defined as follows:

x² + 3x + 3 = 0.

The standard definition of a quadratic function is given as follows:

y = ax² + bx + c.

The solutions are given as follows:

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

Comparing the standard solution to the solution given in this problem, the parameters a and b are given as follows:

The coefficient c is then obtained as follows:

b² - 4ac = -3.

9 - 4c = -3

4c = 12

c = 3.

Hence the function is:

x² + 3x + 3 = 0.

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The area of the trapezium with bases 20cm and 10cm and a height of 16cm is 240 square centimeters.

A trapezium is simply a two-dimensional quadrilateral that has one pair of parallel sides.

The area of a trapezium is expressed as:

Area = 1/2 × ( a + b ) × h

Where a and b are the base measure and h is the height.

From the diagram:

Base a = 20 cm

Base b = 10 cm

Height h 16 cm

Area = ?

Plug the values into the above formula and solve for the area:

Area = 1/2 × ( a + b ) × h

Area = 1/2 × ( 20 + 10 ) × 16

Area = 1/2 × ( 30 ) × 16

Area = 15 × 16

Area = 240 cm²

Therefore, the area of the trapezium is 240 square centimeters.

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Answer: 50$

Step-by-step explanation:

To calculate the amount the college student can spend on fast food in December, we need to find the total amount he can spend in 5 months, given that he wants to spend an average of $50 per month.

Let's start by finding the total amount he can spend in 5 months:

Total amount = 5 x $50 = $250

Now, we can subtract the amount he spent in the other 4 months from the total amount to find out how much he can spend in December:

The amount he can spend in December = Total amount - Amount spent in 4 months

Amount spent in 4 months = $35 + $90 + $15 + $60 = $200

Amount he can spend in December = $250 - $200 = $50

Therefore, the college student can spend $50 on fast food in December to meet his goal of spending an average of $50 per month on fast food for the remaining 5 months of the year.

The system of equations that is represented by the graph include:

D. y = x - 4

   [tex]y=\frac{x-4}{x+2}[/tex]

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of the red line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (0 + 5)/(4 + 1)

Slope (m) = 5/5

Slope (m) = 1

At point (4, 0) and a slope of 1, a linear equation for C can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 0 = 1(x - 4)

y = x - 4

Since the rational function has a y-intercept of (0, -2), it would have a vertical asymptote and the denominator would be undefined at x = 2;

[tex]y=\frac{x-4}{x+2}[/tex]

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The equation of the lines expressed in point-slope form and the average rate of change of the functions indicates that we get;

The point-slope form of the equation of a line can be represented in the following form; y - y₁ = m(x - x₁), where; m is the slope of the line and (x₁, y₁) is a point on the line.

1. (a) The slope of the line parallel to the line; 5·x - 3·y = 10, can be obtained by writing the equation of the line in slope-intercept form as follows;

5·x - 3·y = 10, therefore; y = 5·x/3 - 10/3

The slope of the parallel line is therefore; 5/3

The point-slope form of the equation of the line is therefore;

y - 6 = (5/3)·(x - (-2)) = (5/3)·(x + 2)

y - 6 = (5/3)·(x + 2)

3·y - 18 = 5·x + 10

3·y - 5·x = 10 + 18 = 18

(b) The equation of the line, 2·x + 4·y - 12 = 0, indicates;

The slope of the line is; -2/4 = -1/2

The slope of the perpendicular line = -1/(-1/2) = 2

The equation of the perpendicular line passing through the point (2, -7) in point-slope form is therefore;

y - (-7) = 2·(x - 2)

y + 7 = 2·(x - 2)

(c) The line passing through point (-2, -4), and parallel to y = -3 is the line y = -4

(d) The line passing through point (4, -1), and perpendicular to x = 0 is the line y = -1

2. (a) f(-1) = 2 × (-1)² + (-1) - 1 = 0

f(3) = 2 × (3)² + (3) - 1 = 20

The average is; (f(3) - f(-1))/(3 - (-1)) = 20/4 = 5

(b) f(1) = 150, f(3) = 50

Average = (f(3) - f(1))/(3 - 1)

Average = (150 - 50)/(3 - 1) = 50

(c) f(4) = 4² - 4  

f(4 + h) = (4 + h)² - (4 + h) = (h + 3)·(h + 4)

Average = (f(4 + h) - f(4))/(4 + h - 4) = ((h + 3)·(h + 4) - (4² - 4))/(h)

((h + 3)·(h + 4) - (4² - 4))/(h) = (h² + 7·h + 12 - 12)/h = h + 7

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To determine the minimum sample size required to estimate the proportion of students who bring laptops to campus with a 99% confidence level and a margin of error within five percentage points, we can use the formula:

[tex]n = \frac{(Z^2 \times p \times (1 - p))}{ E^2}[/tex]

Where:

n is the required sample size,

Z is the Z-score corresponding to the desired confidence level,

p is the estimated population proportion (since no prior estimate is available, we use 0.5 as a conservative estimate),

E is the margin of error.

For a 99% confidence level, the Z-score is approximately 2.58 (obtained from the z table).

Plugging in the values:

[tex]n = \frac{(2.58^2 \times 0.5 \times (1 - 0.5))} { (0.05^2)}[/tex]

Simplifying the equation:

n = 663.924

Rounding up to the nearest whole number, the minimum sample size required is approximately 664 students.

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The length of the third side of the triangle, to the nearest hundredth, is approximately 54.75 units.

1. We have a triangle with two known side lengths: 43 and 67 units.

2. The angle included between these sides measures 27 degrees.

3. To find the length of the third side, we can use the Law of Cosines, which states that [tex]c^2 = a^2 + b^2[/tex] - 2ab * cos(C), where c is the third side and C is the included angle.

4. Plugging in the known values, we get [tex]c^2 = 43^2 + 67^2[/tex] - 2 * 43 * 67 * cos(27).

5. Evaluating the expression on the right side, we get [tex]c^2[/tex] ≈ 1849 + 4489 - 2 * 43 * 67 * 0.891007.

6. Simplifying further, we have [tex]c^2[/tex] ≈ 6338 - 5156.898.

7. Calculating [tex]c^2[/tex], we find [tex]c^2[/tex] ≈ 1181.102.

8. Finally, taking the square root of [tex]c^2[/tex], we get c ≈ √1181.102 ≈ 34.32.

9. Rounding to the nearest hundredth, the length of the third side is approximately 34.32 units.

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