3. Determine whether the triangles are similar. If they are, write a similarity statement.
Look at picture for reference
Please show work

Answer:

This is an equation of a parabola.

[tex](y+6)^2=4(x+1)[/tex]

Step-by-step explanation:

A conic section is a curve obtained by the intersection of a plane and a cone. The three major conic sections are parabola, hyperbola and ellipse (the circle is a special type of ellipse).

The standard equations for hyperbolas and ellipses all include x² and y² terms. The standard equation for a parabola includes the square of only one of the two variables.

Therefore, the equation y² - 4x + 12y + 32 = 0 represents a parabola, as there is no x² term.

As the y-variable is squared, the parabola is horizontal (sideways), and has an axis of symmetry parallel to the x-axis.

The conic form of a sideways parabola is:

[tex]\boxed{(y-k)^2=4p(x-h)}[/tex]

where:

To write the given equation in conic form, we need to complete the square for the y-variable.

Rearrange the equation so that the y-terms are on the left side:

[tex]y^2 + 12y = 4x - 32[/tex]

Add the square of half the coefficient of the y-term to both sides of the equation:

[tex]y^2 + 12y+\left(\dfrac{-12}{2}\right)^2 = 4x - 32+\left(\dfrac{-12}{2}\right)^2[/tex]

    [tex]y^2 + 12y+\left(-6\right)^2 = 4x - 32+\left(-6\right)^2[/tex]

         [tex]y^2 + 12y+36 = 4x - 32+36[/tex]

         [tex]y^2 + 12y+36 = 4x +4[/tex]

Factor the perfect square trinomial on the left side of the equation:

[tex](y+6)^2=4x+4[/tex]

Factor out the coefficient of the x-term from the right side of the equation:

[tex](y+6)^2=4(x+1)[/tex]

Therefore, the equation of the given conic section in conic form is:

[tex]\boxed{(y+6)^2=4(x+1)}[/tex]

where:

The conic section of the equation y² - 9x + 12y + 32 = 0 is a parabola

The given equation is

y² - 9x + 12y + 32 = 0

The above equation is an illustration of a parabola equation

The standard form of a parabola is

(x - h)² = 4a(y - k)²

Where

(h, k) is the center

While the general form of the equation is

Ax² + Dx + Ey + F = 0

In this case, the equation y² - 9x + 12y + 32 = 0 takes the general form

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Answer:

When p, the probability of success, is zero in a binomial distribution, the probability of getting exactly k successes in n trials is also zero for all values of k except when k is zero (i.e., when there are no successes).

So, in the case of (0.3)^0, the result would be 1, because any number raised to the power of 0 is equal to 1. Therefore, the probability of getting zero successes in a binomial distribution when the probability of success is 0.3 is 1.

HI Your answer is 42

I have calculated it you can trust me

Well you have marked right in the pic

PLEASE MARK AS BRAINLIEST

The data set has two outliers, namely the points (2, 225) and (8, 75).

Based on the given information about the scatterplot, we can observe that most of the points are grouped closely together and show a slight increase.

There are two points that lie outside of this cluster, specifically (2, 225) and (8, 75).

To determine if these points are outliers, we need to consider their deviation from the general pattern exhibited by the majority of the data points.

If these points deviate significantly from the overall trend, they can be considered outliers.

In this case, since (2, 225) and (8, 75) lie outside of the cluster of closely grouped points and do not follow the general pattern, they can be considered outliers.

These points are noticeably different from the majority of the data points and may have influenced the overall trend of the scatterplot.

The data set has two outliers, namely the points (2, 225) and (8, 75).

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[tex]\qquad\displaystyle \rm \dashrightarrow \: let \: \: Sydney's \: \: age \: \: be \: \: 'y'[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: Devaughn's \: \: age \: \: will \: \: be \: \: 3y[/tex]

Sum up ;

[tex]\qquad\displaystyle \tt \dashrightarrow \: 3y + y = 80[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 4y = 80[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 80 \div 4[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: y = 20[/tex]

So, Sydney's age is 20 years, n that of Devaughn is 20 × 3 = 60 years

Answer:

Sydney= 20, Devaughn= 60

Step-by-step explanation:

Let Sydney's age be 'x'

Devaughn's age = 3 times x = 3x

We Know That

The sum of their ages is 80.

So,

3x + x = 80

4x = 80

If we shift the 4 to the 80 side

x = 80/4

x = 20

So, Sydney's age is 20

Therefore, Devaughn's age =

3x = 3 times x

= 3 times 20

= 60

The domain of f(x) is (0, +∞), and the range is (0, +∞). The graph of the function will have a vertical asymptote at x = 0 and will continuously increase as x approaches positive infinity.

To graph the given logarithmic function f(x) based on the table, we can use the information provided. The table presents pairs of values (x, y), where x represents the input and y represents the output of the function.

From the table, we can observe that the input values (x) are positive and non-zero. This indicates that the domain of the function is x > 0, meaning x is greater than zero. In interval notation, the domain would be written as (0, +∞).

Looking at the output values (y) in the table, we see that they are all positive. This suggests that the range of the function is y > 0, meaning y is greater than zero. In interval notation, the range would be expressed as (0, +∞).

Graphically, the function f(x) is logarithmic and will have a vertical asymptote at x = 0. As x approaches positive infinity, the function increases without bound. The graph starts at y = 125 when x = 1, and it intersects the y-axis at y = 5 when x = 1.5. The graph of the function will resemble a curve that approaches but never touches the x-axis.

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Answer:

f(3r - 1) = -6r + 6

Step-by-step explanation:

To find f(3r - 1), we substitute 3r - 1 for x in the expression for f(x) and simplify:

f(x) = 4 - 2x

f(3r - 1) = 4 - 2(3r - 1)

= 4 - 6r + 2

= -6r + 6

So, f(3r - 1) = -6r + 6.

Answer:

Hope this helps and have a nice day

Step-by-step explanation:

To find the value of x, we can use the formula:

Time = Distance / Speed

Let's calculate the time taken to travel from Q to P and back to Q.

From Q to P:

Distance = 12 km

Speed = 6 km/h

Time taken from Q to P = Distance / Speed = 12 km / 6 km/h = 2 hours

From P to Q:

Distance = 12 km

Speed = (6 + x) km/h

Time taken from P to Q = Distance / Speed = 12 km / (6 + x) km/h

Given that the total time taken for the round trip is 3 hours 20 minutes, we can convert it to hours:

Total time = 3 hours + (20 minutes / 60) hours = 3 + (1/3) hours = 10/3 hours

According to the problem, the total time is the sum of the time from Q to P and from P to Q:

Total time = Time taken from Q to P + Time taken from P to Q

Substituting the values:

10/3 hours = 2 hours + 12 km / (6 + x) km/h

Simplifying the equation:

10/3 = 2 + 12 / (6 + x)

Multiply both sides by (6 + x) to eliminate the denominator:

10(6 + x) = 2(6 + x) + 12

60 + 10x = 12 + 2x + 12

Collecting like terms:

8x = 24

Dividing both sides by 8:

x = 3

Therefore, the value of x is 3.

Answer:

x = 3

Step-by-step explanation:

speed  = distance / time

time = distance / speed

Total time from P to Q to P:

T = 3h 20min

P to Q :

s = 6 km/h

d = 12 km

t = d/s

= 12/6

t = 2 h

time remaining t₁ = T - t

= 3h 20min - 2h

=  1 hr 20 min

= 60 + 20 min

= 80 min

t₁ = 80/60 hr

Q to P:

d₁ = 12km

t₁ = 80/60 hr

s₁ = d/t₁

[tex]= \frac{12}{\frac{80}{60} }\\ \\= \frac{12*60}{80}[/tex]

= 9

s₁ = 9 km/h

From question, s₁ = (6 + x)km/h

⇒ 6 + x = 9

⇒ x = 3

Answer:

Step-by-step explanation:

The quadratic formula is y=ax^2+bx+c

If we move everything to the left side of the equation,

-6x^2=-9x+7 becomes

-6x^2+9x-7=0

a=-6, b=9, c=-7, so the third answer choice

Answer:

Step-by-step explanation:

To find the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis, we can use the method of cylindrical shells.

The formula for the volume of a solid obtained by rotating a region bounded by the graph of a function f(x), the x-axis, x = a, and x = b about the y-axis is given by:

V = 2π ∫[a, b] x * f(x) dx

In this case, the function f(x) = (x - 4)^3, and the bounds of integration are a = 0 and b = 5.

Substituting these values into the formula, we have:

V = 2π ∫[0, 5] x * (x - 4)^3 dx

To evaluate this integral, we can expand the cubic term and then integrate:

V = 2π ∫[0, 5] x * (x^3 - 12x^2 + 48x - 64) dx

V = 2π ∫[0, 5] (x^4 - 12x^3 + 48x^2 - 64x) dx

Integrating each term separately:

V = 2π [1/5 x^5 - 3x^4 + 16x^3 - 32x^2] evaluated from 0 to 5

Now we can substitute the bounds of integration:

V = 2π [(1/5 * 5^5 - 3 * 5^4 + 16 * 5^3 - 32 * 5^2) - (1/5 * 0^5 - 3 * 0^4 + 16 * 0^3 - 32 * 0^2)]

Simplifying:

V = 2π [(1/5 * 3125) - 0]

V = 2π * (625/5)

V = 2π * 125

V = 250π

Therefore, the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis is 250π cubic units.

Answer:

Step-by-step explanation:

The correct answer is A. A rectangle the same size as the base.

When a rectangular pyramid is sliced parallel to the base, the resulting cross-section is a rectangle that is the same size as the base. The parallel slicing ensures that the cross-section maintains the same dimensions as the base of the pyramid. Therefore, option A, a rectangle the same size as the base, represents the cross-section.

Step-by-step explanation:

a linear relationship or function is described in general as

y = f(x) = ax + b

Because the variable term has the variable x only with the exponent 1, this makes this a straight line - hence the name "linear".

here f(x) is P(x) :

P(x) = ax + b

now we are using both given points (ordered pairs) to calculate a and b :

20 = a×170 + b

2820 = a×220 + b

to eliminate first one variable we subtract equation 1 from equation 2 :

2800 = a×50

a = 2800/50 = 280/5 = 56

now, we use that in any of the 2 original equations to get b :

20 = 56×170 + b

b = 20 - 56×170 = 20 - 9520 = -9500

so,

P(x) = 56x - 9500

Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C).

The given functions are f(x) = 3x² and g(x) = -2².

To understand the difference between their graphs, let's examine the characteristics of each function individually:

Function f(x) = 3x²:

The coefficient of x² is positive (3), indicating an upward-opening parabola.

The graph of f(x) will be symmetric with respect to the y-axis, as any change in x will result in the same y-value due to the squaring of x.

The vertex of the parabola will be at the origin (0, 0) since there are no additional terms affecting the position of the graph.

Function g(x) = -2²:

The coefficient of x² is negative (-2), indicating a downward-opening parabola.

The negative sign will reflect the graph of f(x) across the x-axis, resulting in a vertical flip.

The vertex of the parabola will also be at the origin (0, 0) due to the absence of additional terms.

Comparing the characteristics of the two functions, we can conclude that the graph of g(x) = -2² is a reflection of the graph of f(x) = 3x² over the x-axis (option C). This means that g(x) is obtained by taking the graph of f(x) and flipping it vertically. The reflection occurs over the x-axis, causing the parabola to open downward instead of upward.

Therefore, the correct answer is option C: g(x) is a reflection of f(x) over the x-axis.

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There are 70 ways a person can order a two-course meal from the given restaurant.

To determine the number of ways a person can order a two-course meal from a restaurant that offers 10 appetizers and 7 main courses, we can use the concept of combinations.

First, we need to select one appetizer from the 10 available options.

This can be done in 10 different ways.

Next, we need to select one main course from the 7 available options. This can be done in 7 different ways.

Since the two courses are independent choices, we can multiply the number of options for each course to find the total number of combinations.

Therefore, the number of ways a person can order a two-course meal is 10 [tex]\times[/tex] 7 = 70.

So, there are 70 ways a person can order a two-course meal from the given restaurant.

It's important to note that this calculation assumes that a person can choose any combination of appetizer and main course.

If there are any restrictions or limitations on the choices, the number of combinations may vary.

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The expression (3√2) / (3√6) with a rationalized denominator is 3√9 / 6. Option C is the correct answer.

To rationalize the denominator in the expression (3√2) / (3√6), we can multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of √6 is -√6, so we multiply the expression by (-√6) / (-√6):

(3√2 / 3√6) * (-√6 / -√6)

This simplifies to:

-3√12 / (-3√36)

Further simplifying, we have:

-3√12 / (-3 * 6)

-3√12 / -18

Finally, we can cancel out the common factor of 3:

- 3√9 / - 6.

Simplifying further, we get:

3√9 / 6.

Option C is the correct answer.

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Answer:

Part A:

The correct expression for how many pencils Ms. Florinda has left after giving them out to her students is:

A. 100 - 2 × (22 - 19)

Part B:

To determine whether Ms. Florinda has enough pencils to give each of her students 2, we can calculate the total number of pencils needed. The total number of students is the sum of the students in both classes, which is 22 + 19 = 41.

If each student needs 2 pencils, then the total number of pencils needed is 2 × 41 = 82.

Comparing this with the initial number of pencils Ms. Florinda bought (100), we can see that she has more than enough pencils to give each of her students 2.

Yes, she has enough pencils to give each of her students 2.

She has 18 more than she needs.

Answer:

The volume of the cylinder is 628.318530718

Step-by-step explanation:

The formula used to find the volume of a cylinder (v) is [tex]v = \pi r^2h[/tex], where r = radius and h = height. As the question says to keep the answer exact, we will be using pi as opposed to 3.14.

The radius is 5, and the height is 8. Plug these values into the equation and solve:

[tex]v =\pi *5^2*8[/tex]

[tex]v = 628.318530718[/tex]

So, the exact volume of the cylinder is 628.318530718. Rounded is 628.32

Answer:

200π or 628

Step-by-step explanation:

Note: your picture is not clear so I am assuming the height to be 8.

r = 5

h = 8

Volume = πr²h

= π * 5² * 8

= (25*8) π

= 200π

= 200*3.14

= 628

The computed mean of 52.4 and the actual mean of 52.7 suggest a close relationship in terms of central tendency.

A computed mean is a statistical measure calculated by summing up a set of values and dividing by the number of observations. In this case, the computed mean of 52.4 implies that when the values are averaged, the result is 52.4.

The actual mean of 52.7 refers to the true average of the population or data set being analyzed. Since it is higher than the computed mean, it indicates that the sample used for computation might have slightly underestimated the true population mean.

However, the difference between the computed mean and the actual mean is relatively small, with only a 0.3 unit discrepancy.

Given the proximity of these two values, it suggests that the computed mean is a reasonably accurate estimate of the actual mean.

However, it's important to note that without additional information, such as the sample size or the variability of the data, it is difficult to draw definitive conclusions about the relationship between the computed mean and the actual mean.

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The equivalent ratio of the corresponding sides and the triangle proportionality theorem indicates that the similar triangles are;

1. ΔAJK ~ ΔSWY according to the SAS similarity postulate

2. ΔLMN ~ ΔLPQ according to the AA similarity postulate

3. ΔPQN ~ ΔLMN

LM = 12, QP = 8

4. ΔLMK~ΔLNJ

NL = 21, ML = 14

Similar triangles are triangles that have the same shape but may have different sizes.

1. The ratio of corresponding sides between the two triangles circumscribing the congruent included angle are;

24/16 = 3/2

18/12 = 3/2

The ratio of each of the two sides in the triangle ΔAJK to the corresponding sides in the triangle ΔSWY are equivalent and the included angle, therefore, the triangles ΔAJK and ΔSWY are similar according to the SAS similarity rule.

2. The ratio of the corresponding sides in each of the triangles are;

MN/LN = 8/10 = 4/5

PQ/LQ = 12/(10 + 5) = 12/15 = 4/5

The triangle proportionality theorem indicates that the side MN and PQ are parallel, therefore, the angles ∠LMN ≅ ∠LPQ and ∠LNM ≅ ∠LQP, which indicates that the triangles ΔLMN and ΔLPQ are similar according to the Angle-Angle AA similarity rule

3. The alternate interior angles theorem indicates;

Angles ∠PQN ≅ ∠LMN and ∠MLN ≅ ∠NPQ, therefore;

ΔPQN ~ ΔLMN by the AA similarity postulate

LM/QP = (x + 3)/(x - 1) = 18/12

12·x + 36 = 18·x - 18

18·x - 12·x = 36 + 18 = 54

6·x = 54

x = 54/6 = 9

LM = 9 + 3 = 12

QP = x - 1

QP = 9 - 1 = 8

4. The similar triangles are; ΔLMK and ΔLNJ

ΔLMK ~ ΔLNJ by AA similarity postulate

ML/NL = (6·x + 2)/(6·x + 2 + (x + 5)) = (6·x + 2)/((7·x + 7)

ML/NL = LK/LJ = (24 - 8)/24

(24 - 8)/24 = (6·x + 2)/((7·x + 7)

16/24 = (6·x + 2)/(7·x + 7)

16 × (7·x + 7) = 24 × (6·x + 2)

112·x + 112 = 144·x + 48

144·x - 112·x = 32·x = 112 - 48 = 64

x = 64/32 = 2

ML = 6 × 2 + 2 = 14

NL = 7 × 2 + 7 = 21

MN = 2 + 5 = 7

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The z-score for Brazil is given as follows:

Z = 0.87.

The z-score formula is defined as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

In which:

The parameters for this problem are given as follows:

[tex]X = 6.24, \mu = 4.8, \sigma = 1.66[/tex]

Hence the z-score for Brazil is given as follows:

Z = (6.24 - 4.8)/1.66

Z = 0.87.

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The box plot that best represents the numerical data is: A. A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.

In order to determine the five-number summary for the survey, we would arrange the data set in an ascending order:

4.5,6,6,6.5,7,7,7.5,8,8.5,9,9.5,10,11

Based on the information provided about the list of shoe sizes for a group of 13 people, we would use a graphical method (box plot) to determine the five-number summary for the given data set as follows:

Minimum (Min) = 4.5.

First quartile (Q₁) = 6.25.

Median (Med) = 7.5.

Third quartile (Q₃) = 9.25.

Maximum (Max) = 11.

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The proposal is as follows

Part III  -  The distance between buoys A and B is 12.8 kilometers.

Part IV  -  The length of the other two triangle legs are 10.2 kilometers and 8.4 kilometers.

Part V  - The total length of the race course is 31.4 kilometers.

Part VIII  - The winner's speed during last year's race was 10.8 knots.

See the proposal attached.

Dear Sponsor,

I'm seeking sponsorship for the San Francisco sailboat race.

With a proven track record and the determination to win, your investment of $5,500 covers travel costs and potential hazards.

By associating your brand with a winning sailor, you'll gain significant exposure to thousands of spectators. Join me in this thrilling race for success.

Sincerely,

[Your Name]

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What is needed to be congruent to show that ABC=AXYZ is AC ≅ XZ. option D

Given that in ΔABC and ΔXYZ, ∠X ≅ ∠A and ∠Z ≅ ∠C.

We are to select the correct condition that we will need to show that the triangles ABC and XYZ are congruent to each other by ASA rule..

ASA Congruence Theorem: Two triangles are said to be congruent if two angles and the side lying between them of one triangle are congruent to the corresponding two angles and the side between them of the second triangle.

In ΔABC, side between ∠A and ∠C is AC,

in ΔXYZ, side between ∠X and ∠Z is XZ.

Therefore, for the triangles to be congruent by ASA rule, we must have AC ≅ XZ.

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The tax charged on Omari's monthly earning of Ksh 24,200 is Ksh 3,340.

To calculate the tax charged on Omari's monthly earning, we need to consider the tax brackets and rates applicable in the specific tax system or country. Since you haven't specified a particular tax system, I will provide a general explanation.

Assuming we have a simplified progressive tax system with three tax brackets:

For the first tax bracket, let's say income up to Ksh 10,000 is taxed at a rate of 10%.

For the second tax bracket, income between Ksh 10,001 and Ksh 20,000 is taxed at a rate of 15%.

For the third tax bracket, income above Ksh 20,000 is taxed at a rate of 20%.

To calculate the tax charged on Omari's monthly earning of Ksh 24,200, we can divide it into the respective tax brackets:

Ksh 10,000 falls in the first tax bracket. So, the tax for this portion is 10% of Ksh 10,000, which is Ksh 1,000.

Ksh 20,000 - Ksh 10,000 = Ksh 10,000 falls in the second tax bracket. The tax for this portion is 15% of Ksh 10,000, which is Ksh 1,500.

The remaining amount, Ksh 24,200 - Ksh 20,000 = Ksh 4,200, falls in the third tax bracket. The tax for this portion is 20% of Ksh 4,200, which is Ksh 840.

Now, we can sum up the taxes for each bracket:

Total Tax = Tax in the first bracket + Tax in the second bracket + Tax in the third bracket

Total Tax = Ksh 1,000 + Ksh 1,500 + Ksh 840

Total Tax = Ksh 3,340

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The graph of the function and its inverse is added as an attachment

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

f(x) = 3 + 2ln(x)

Express as an equation

So, we have

y = 3 + 2ln(x)

Swap x and  y in the above equation

x = 3 + 2ln(y)

Next, we have

2ln(y) = x - 3

Divide by 2

ln(y) = (x - 3)/2

Take the exponent of both sides

[tex]y = e^{\frac{x - 3}{2}}[/tex]

Next, we plot the graphs

The graph of the functions is added as an attachment

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Answer: [tex](-\infty, 0) \cup (0, \infty)[/tex]

Step-by-step explanation:

The graph of [tex]\frac{\sin x}{x}[/tex] is continuous for all real [tex]x[/tex] except [tex]x=0[/tex], and multiplying this by [tex]1/3[/tex] does not change this.

The given expression, 10f - 5f + 8 + 6g + 4, simplifies to 5f + 12 + 6g when like terms are combined.

To simplify the expression 10f - 5f + 8 + 6g + 4, we can combine like terms by adding or subtracting coefficients that have the same variables:

10f - 5f + 8 + 6g + 4

Combining the terms with 'f', we have:

(10f - 5f) + 8 + 6g + 4

This simplifies to:

5f + 8 + 6g + 4

Next, we can combine the constant terms:

8 + 4 = 12

Thus, the simplified expression is:

5f + 12 + 6g

This expression is equivalent to 10f - 5f + 8 + 6g + 4.

In summary, the expression 10f - 5f + 8 + 6g + 4 simplifies to 5f + 12 + 6g after combining like terms.

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Answer:

D The base is 7 units and the height is 3 units.

Step-by-step explanation:

The answer is d I counted the width/base then the height/length and found answer.

Answer:

[tex]\dfrac{\sqrt{2}-\sqrt{6} }{4} }[/tex]

Step-by-step explanation:

Find the exact value of cos(105°).

The method I am about to show you will allow you to complete this problem without a calculator. Although, memorizing the trigonometric identities and the unit circle is required.    

We have,

[tex]\cos(105\°)[/tex]

Using the angle sum identity for cosine.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Angle Sum Identity for Cosine}}\\\\\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)\end{array}\right}[/tex]

Split the given angle, in degrees, into two angles. Preferably two angles we can recognize on the unit circle.

[tex]105\textdegree=45\textdegree+60\textdegree\\\\\\\therefore \cos(105\textdegree)=\cos(45\textdegree+60\textdegree)[/tex]

Now applying the identity.

[tex]\cos(45\textdegree+60\textdegree)\\\\\\\Longrightarrow \cos(45\textdegree+60\textdegree)=\cos(45\textdegree)\cos(60\textdegree)-\sin(45\textdegree)\sin(60\textdegree)[/tex]

Now utilizing the unit circle.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{From the Unit Circle:}}\\\\\cos(45\textdegree)=\dfrac{\sqrt{2} }{2}\\\\\cos(60\textdegree)=\dfrac{1}{2}\\\\\sin(45\textdegree)=\dfrac{\sqrt{2} }{2}\\\\\sin(60\textdegree)=\dfrac{\sqrt{3} }{2} \end{array}\right}[/tex]

[tex]\cos(45\textdegree)\cos(60\textdegree)-\sin(45\textdegree)\sin(60\textdegree)\\\\\\\Longrightarrow \Big(\dfrac{\sqrt{2} }{2}\Big)\Big(\dfrac{1 }{2}\Big)-\Big(\dfrac{\sqrt{2} }{2}\Big)(\dfrac{\sqrt{3} }{2}\Big)[/tex]

Now simplifying...

[tex]\Big(\dfrac{\sqrt{2} }{2}\Big)\Big(\dfrac{1 }{2}\Big)-\Big(\dfrac{\sqrt{2} }{2}\Big)(\dfrac{\sqrt{3} }{2}\Big)\\\\\\\Longrightarrow \Big(\dfrac{\sqrt{2} }{4} \Big)-\Big(\dfrac{\sqrt{6} }{4} \Big)\\\\\\\therefore \cos(105\textdegree)= \boxed{\boxed{\frac{\sqrt{2}-\sqrt{6} }{4} }}[/tex]