A 52-card deck contains 13 cards from each of the four suits: clubs ♣, diamonds ♦, hearts ♥, and spades ♠. You deal four cards without replacement from a well-shuffled deck so that you are equally likely to deal any four cards.


What is the probability that all four cards are clubs?


13/52 ⋅ 12/51 ⋅ 11/50 ⋅ 10/49 ≈0.0026



13/52 ⋅ 12/52 ⋅ 11/52 ⋅ 10/52 ≈0.0023



1/4 because 1/4 of the cards are clubs

Answer:

Yes, a sinusoidal function is a great way to model temperatures over a 24-hour period because the pattern of temperature changes tends to be cyclic.

A sinusoidal function can be written in the general form:

D(t) = A sin(B(t - C)) + D

where:

- A is the amplitude (half the range of the temperature changes)

- B is the frequency of the cycle (which would be `2π/24` in this case because the temperature completes a full cycle every 24 hours)

- C is the horizontal shift (which is determined by the fact that the minimum temperature occurs at 6 AM)

- D is the vertical shift (which is the average of the maximum and minimum temperature)

Given the information you've provided, let's fill in the specifics:

- The high temperature for the day is 95 degrees.

- The low temperature is 75 degrees at 6 AM.

The amplitude, A, is half the range of temperature changes. It's the difference between the high and the low temperature divided by 2:

A = (95 - 75) / 2 = 10

The frequency, B, is `2π/24` because the temperature completes a full cycle every 24 hours.

The horizontal shift, C, is determined by the fact that the minimum temperature occurs at 6 AM. The sine function hits its minimum halfway through its period, so we want to shift the function to the right by 6 hours to make this happen. In our case, this means C = 6.

The vertical shift, D, is the average of the maximum and minimum temperature:

D = (95 + 75) / 2 = 85

So the equation for the temperature, D, in terms of t (the number of hours since midnight) is:

D(t) = 10 sin((2π/24) * (t - 6)) + 85

This equation represents a sinusoidal function that models the temperature over a day given the information provided.

The values x = -5 and x = 3 make the second expression undefined. The correct answers are:

x = -5

x = 3

x= -3

To determine the values for which the given expressions are undefined, we need to find the values that make the denominators equal to zero.

First expression: [tex]\frac{3x}{(x^2 - 9)}[/tex]

For this expression, the denominator is (x^2 - 9). It will be undefined when the denominator equals zero:

x^2 - 9 = 0

Factoring the equation, we have:

(x - 3)(x + 3) = 0

Setting each factor equal to zero, we get:

x - 3 = 0 --> x = 3

x + 3 = 0 --> x = -3

So, the values x = 3 and x = -3 make the first expression undefined.

Second expression: [tex]\frac{(x + 4)}{(x^2 + 2x - 15)}[/tex]

For this expression, the denominator is (x^2 + 2x - 15). It will be undefined when the denominator equals zero:

x^2 + 2x - 15 = 0

Factoring the equation, we have:

(x + 5)(x - 3) = 0

Setting each factor equal to zero, we get:

x + 5 = 0 --> x = -5

x - 3 = 0 --> x = 3

So, The second expression is ambiguous because x = -5 and x = 3.

Consequently, the right responses are x = -5, x = 3 and x= -3.

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Answer:

[tex]x = 5.3 \: or \: 5 \frac{1}{3} [/tex]

Step-by-step explanation:

[tex]2 \times 2 + 3x - 20 = 0 \: \: \: \: \: \: 4 + 3x - 20 = 0 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:3x = 20 - 4 = 16 \: \: \: \: \: \: \: \: 3x = 16 \: \: divide \: both \: side \: by \: 3 = \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: x = 5.3[/tex]

Answer:

To find the length of AB using the segment addition postulate , we need to add the lengths of segments AC and CB.

AC + CB = AB

Substituting the given lengths:

2x-3 + 24 = 5x+6

Simplifying and solving for x:

21 = 3x

x = 7

Now that we know x, we can substitute it back into the expression for AB:

AB = 2x-3 + 24 = 2(7)-3 + 24 = 14-3+24 = 35

Therefore, the length of AB is 35.

Step-by-step explanation:

The correct answer is By finding the quotient of the bases to be one third and canceling common factors. Option D.

The Quotient of Powers Property states that when dividing two powers with the same base, you can subtract the exponents. In the given expression, we have three to the fourth power divided by nine.

To simplify this expression using the Quotient of Powers Property, we first need to recognize that nine can be written as three squared, since 3 multiplied by itself gives 9.

So, we have (3^4) / (3^2). According to the Quotient of Powers Property, we subtract the exponents: 4 - 2.

This gives us 3^(4-2), which simplifies to 3^2. Therefore, the expression three to the fourth power all over nine equals three squared.

It states that we find the quotient of the bases to be one third and cancel common factors. In this case, the bases are 3 and 3, and their quotient is indeed one third. Additionally, there are no common factors that can be canceled, as the expression does not contain any variables or additional terms.

Therefore, By finding the quotient of the bases to be one third and canceling common factors. accurately describes the steps involved in simplifying the expression using the Quotient of Powers Property.

We find the quotient of the bases (one third) and cancel common factors (which is not applicable in this case). Option D is correct.

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Note the complete question is

Explain how the Quotient of Powers Property was used to simplify this expression. (1 point)

Three to the fourth power all over nine equals three squared

A.) By simplifying 9 to 32 to make both powers base three and adding the exponents

B.) By simplifying 9 to 32 to make both powers base three and subtracting the exponents

C.) By finding the quotient of the bases to be one third and simplifying the expression

D.) By finding the quotient of the bases to be one third and cancelling common factors

Approximately 2 tissues were used by each groom's guest at the wedding.

The calculation is as follows:

180 tissues ÷ 90 guests = 2 tissues per guest.

To determine how many tissues each groom's guest used, we need to find the average number of tissues per guest. We start by adding up the number of tissues used by the groom's guests, which is 180.

Then, we divide this total by the number of groom's guests, which is 90. This division gives us an average of 2 tissues per guest.

By dividing the total number of tissues used by the total number of guests, we can find the average number of tissues per guest. In this case, each groom's guest used approximately 2 tissues.

It's important to note that this calculation assumes an equal distribution of tissues among all the groom's guests.

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The value of m<RQS as required to be determined in the task content is; 70°.

It follows from the task content that the measure of angle RQS is to be determined as required.

Recall, the measure of the central angle subtended by an arc is twice that which it subtends at any point on the circumference.

Therefore, m<RPS = 2 • m<RQS.

m<RQS = 140°/2

m<RQS = 70°.

Ultimately, the measure of angle RQS are; 70°.

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Answer:

a) x² - x - 12 = 0

Step-by-step explanation:

We have equation A = equation B

⇒ x² - 9 = x + 3

⇒ x² - 9 - x - 3 = 0

⇒ x² - x - 12 = 0

The scale factor used to create the image of polygon ABCD is 4.

To determine the scale factor, we need to compare the corresponding side lengths of the original polygon ABCD and the image polygon ABCD. Let's denote the scale factor as k.

Original polygon ABCD:

Side AB: length = 3 - 1 = 2

Side BC: length = -2 - (-1) = -1

Side CD: length = 1 - 3 = -2

Side DA: length = -2 - (-1) = -1

Image polygon ABCD:

Side AB: length = 12 - 4 = 8

Side BC: length = -3 - (-4) = 1

Side CD: length = 4 - 12 = -8

Side DA: length = -3 - (-4) = 1

Comparing the corresponding side lengths, we can set up the following equations:

k * 2 = 8 (for side AB)

k * (-1) = 1 (for side BC)

k * (-2) = -8 (for side CD)

k * (-1) = 1 (for side DA)

From the equations, we can see that k = 4 satisfies all of them.

Therefore, the scale factor used to create the image of polygon ABCD is 4.

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The actual duration of eruption for x = 12.03 is 1.97 minutes, so the residual is 1.97 - 3.8431 = -1.8731 minutes.

The value of the linear correlation coefficient, also known as the Pearson correlation coefficient, measures the strength and direction of the linear relationship between two variables.

In this case, it represents the correlation between the time between eruptions and the duration of the eruption. To calculate the linear correlation coefficient, we can use the given data. The linear correlation coefficient is 0.8404.

The coefficient of determination, denoted as R-squared, represents the proportion of the variance in the dependent variable (duration of eruption) that can be explained by the independent variable (time between eruptions).

It is calculated by squaring the linear correlation coefficient. In this case, the coefficient of determination is 0.7055.

The regression line represents the best-fit line that approximates the relationship between the independent and dependent variables.

It can be expressed in the form of y = mx + b, where y represents the predicted duration of the eruption, x represents the time between eruptions, m represents the slope of the line, and b represents the y-intercept.

To determine the regression line, we can perform linear regression analysis using the given data. The regression line is y = 0.1608x + 1.8305.

The predicted duration of an eruption can be calculated by substituting the given time between eruptions value into the regression line equation. For x = 12.03 minutes, the predicted duration of an eruption is y = 0.1608 x 12.03 + 1.8305 = 3.8431 minutes.

The residual for x = 12.03 is the difference between the actual duration of eruption and the predicted duration. It can be calculated by subtracting the predicted value from the actual value. The actual duration of eruption for x = 12.03 is 1.97 minutes, so the residual is 1.97 - 3.8431 = -1.8731 minutes.

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The diagonal distance from home plate to third base is approximately √16,200 feet.

The correct answers are:

B. 90 feet

C. √16,200 feet

D. 180 feet.

In baseball, the bases are arranged in a square shape.

The distance between each base is 90 feet.

Therefore, the correct answer for the distance a player at first base has to throw to a player at third base is 90 feet (option B).

To find the diagonal distance from home plate to third base, we can use the Pythagorean theorem.

Since the area of the baseball field is a square, the diagonal distance represents the hypotenuse of a right triangles.

The two legs of the right triangle are the sides of the square, which are 90 feet each.

Using the Pythagorean theorem [tex](a^2 + b^2 = c^2),[/tex] we can calculate the diagonal distance:

a = b = 90 feet

[tex]c^2 = 90^2 + 90^2[/tex]

[tex]c^2 = 8,100 + 8,100[/tex]

[tex]c^2 = 16,200[/tex]

c = √16,200 feet (option C)

Therefore, the diagonal distance from home plate to third base is approximately √16,200 feet.

The options A, √180 feet, and 180 feet are incorrect because they do not represent the correct distances in the given scenario.

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Answer:

8

Step-by-step explanation:

Substitute the values in the expression, we have:

[tex]\displaystyle{|-1-3|+4}[/tex]

Evaluate:

[tex]\displaystyle{|-4|+4}[/tex]

Any real numbers in the absolute sign will always be evaluated as positive values. Thus:

[tex]\displaystyle{|-4|+4 = 4+4}\\\\\displaystyle{=8}[/tex]

Hence, the answer is 8. A quick note that z-value is not used due to lack of z-term in the expression.

The function g(x) = 4x² - 28x + 49 can be rewritten as g(x) = 4(x - 7/2)² - 147 after completing the square.

To complete the square for the function g(x) = 4x² - 28x + 49, we follow these steps:

Step 1: Divide the coefficient of x by 2 and square the result.

  (Coefficient of x) / 2 = -28/2 = -14

  (-14)² = 196

Step 2: Add and subtract the value obtained in Step 1 inside the parentheses.

  g(x) = 4x² - 28x + 49

  = 4x² - 28x + 196 - 196 + 49

Step 3: Rearrange the terms and factor the perfect square trinomial.

  g(x) = (4x² - 28x + 196) - 196 + 49

  = 4(x² - 7x + 49) - 147

  = 4(x² - 7x + 49) - 147

Step 4: Write the perfect square trinomial as the square of a binomial.

  g(x) = 4(x - 7/2)² - 147

Therefore, the function g(x) = 4x² - 28x + 49 can be rewritten as g(x) = 4(x - 7/2)² - 147 after completing the square.

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The probable question may be:

Rewrite the function by completing the square.

g(x)=4x²-28x +49

g(x)= ____  (x+___ )²+____.

Answer:

550 gallons

Step-by-step explanation:

Let [tex]x[/tex] be the number of gallons for the 90% antifreeze solution and [tex]x+100[/tex] be the total number of gallons that will contain 80% antifreeze solution:

[tex]\displaystyle \frac{0.90x+0.25(100)}{x+100}=0.80\\\\0.90x+25=0.80x+80\\\\0.10x+25=80\\\\0.10x=55\\\\x=550[/tex]

Therefore, you would need 550 gallons of the 90% antifreeze solution.

Answer:

27)  34.29 in²

28)  If I get an A, then I studied for my final.

Step-by-step explanation:

To calculate the area of the trapezoid, we need to find its perpendicular height.

As the given diagram shows an isosceles trapezoid (since the non-parallel sides (the legs) are of equal length), we can use Pythagoras Theorem to calculate the perpendicular height.

Identify the right triangle formed by drawing the perpendicular height from the vertex of the bottom base to the top base (this has been done for you in the given diagram).

As the two base angles of an isosceles trapezoid are always congruent, the base of the right triangle is half the difference between the lengths of the parallel bases, which is (8 - 6)/2 = 1 inch.

The hypotenuse of the right triangle is the leg of the trapezoid, which is 5 inches.

Use Pythagoras Theorem to find the perpendicular height (the length of the other leg):

[tex]h^2+1^2=5^2[/tex]

[tex]h^2+1=25[/tex]

      [tex]h^2=24[/tex]

        [tex]h=\sqrt{24}[/tex]

        [tex]h=2\sqrt{6}[/tex]

Now we have found the height of the trapezoid, we can use the following formula to calculate its area:

[tex]\boxed{\begin{minipage}{7 cm}\underline{Area of a trapezoid}\\\\$A=\dfrac{1}{2}(a+b)h$\\\\where:\\ \phantom{ww}$\bullet$ $A$ is the area.\\ \phantom{ww}$\bullet$ $a$ and $b$ are the parallel sides (bases).\\\phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

The values to substitute into the area formula are:

Substituting these values into the formula we get:

[tex]A=\dfrac{1}{2}(8+6) \cdot 2\sqrt{6}[/tex]

[tex]A=\dfrac{1}{2}(14) \cdot 2\sqrt{6}[/tex]

[tex]A=7\cdot 2\sqrt{6}[/tex]

[tex]A=14\sqrt{6}[/tex]

[tex]A=34.29\; \sf in^2\;(nearest\;hundredth)[/tex]

Therefore, the area of the isosceles trapezoid is 34.29 in², rounded to the nearest hundredth.

[tex]\hrulefill[/tex]

Given conditional statement:

The hypothesis is "I studied for my final", and the conclusion is "I will get an A".

The converse of a conditional statement involves switching the hypothesis ("if" part) and the conclusion ("then" part) of the original statement.

Therefore, the converse of the statement would be:

The surface area and volume of the composite solid is are 1720ft² and 3563.33 ft³ respectively.

The area occupied by a three-dimensional object by its outer surface is called the surface area.

The surface area of the solid = lateral area of pyramid + surface area of cuboid

lateral area of pyramid = 4 × 1/2 bh

= 4 × 1/2 × 10× 12

= 120×2 = 240 ft²

Surface area of the cuboid = 2( 100+ 320+ 320)

= 2( 740)

= 1480 ft²

Surface area of the composite solid = 240 + 1480

= 1720 ft²

Volume of the composite solid = volume of cuboid + volume of pyramid

volume of cuboid = 10×10×32 = 3200ft²

volume of pyramid = 1/3base area × height

height of the pyramid is calculated as;

diagonal of base = √ 10²+10²

= √200

= 14.14

h² = 13²-7.07²

h² = 169 - 49.98

h² = 119.02

h = 10.9 ft

Volume of pyramid = 1/3 × 100 × 10.9

= 363.33 ft³

Volume of the composite solid = 3200+363.33

= 3563.33 ft³

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Answer:

hope you understand it and please follow me

We conclude that there is no valid domain for the given function g(x, y) = sin^-1(x² + y² - 3). Thus, the concept of circles with radii does not apply in this case.

To determine the domain of the function g(x, y) = sin^-1(x² + y² - 3), we need to examine the range of the arcsine function. The arcsine function, [tex]sin^{(-1)[/tex](z), is defined for values of z between -1 and 1, inclusive. Therefore, for the given function, we have:

-1 ≤ x² + y² - 3 ≤ 1

Rearranging the inequality, we get:

-4 ≤ x² + y² ≤ -2

Now, let's analyze the inequalities separately:

x² + y² ≤ -2:

This inequality is not possible since the sum of squares of two non-negative numbers (x² and y²) cannot be negative. Therefore, there are no points that satisfy this inequality.

x² + y² ≤ -4:

Similarly, this inequality is also not possible since the sum of squares of two non-negative numbers cannot be less than or equal to -4. Therefore, there are no points that satisfy this inequality either.

Based on the analysis, we conclude that there is no valid domain for the given function g(x, y) = sin^-1(x² + y² - 3). Thus, the concept of circles with radii does not apply in this case.

It's important to note that the arcsine function has a restricted range of -π/2 to π/2, and for a valid domain, the input of the arcsine function must be within the range of -1 to 1. In this particular case, the given expression x² + y² - 3 exceeds the range of the arcsine function, resulting in no valid domain.

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Answer:

[tex]\text{a.} \quad m\angle NLM=93^{\circ}[/tex]

[tex]\text{c.} \quad m\angle FHG=31^{\circ}[/tex]

Step-by-step explanation:

The inscribed angle in the given circle is ∠NLM.

The intercepted arc in the given circle is arc NM = 186°.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc.

Therefore:

[tex]m\angle NLM=\dfrac{1}{2}\overset{\frown}{NM}[/tex]

[tex]m\angle NLM=\dfrac{1}{2} \cdot 186^{\circ}[/tex]

[tex]\boxed{m\angle NLM=93^{\circ}}[/tex]

[tex]\hrulefill[/tex]

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore:

[tex]m\angle HFG=\dfrac{1}{2}\overset{\frown}{HG}[/tex]

[tex]m\angle HFG=\dfrac{1}{2}\cdot 118^{\circ}[/tex]

[tex]m\angle HFG=59^{\circ}[/tex]

As line segment FH passes through the center of the circle, FH is the diameter of the circle. Since the angle at the circumference in a semicircle is a right angle, then:

[tex]m\angle FGH = 90^{\circ}[/tex]

The interior angles of a triangle sum to 180°. Therefore:

[tex]m\angle FHG + m\angle HFG + m\angle FGH =180^{\circ}[/tex]

[tex]m\angle FHG + 59^{\circ} + 90^{\circ} =180^{\circ}[/tex]

[tex]m\angle FHG +149^{\circ} =180^{\circ}[/tex]

[tex]\boxed{m\angle FHG =31^{\circ}}[/tex]

The number line that represents the solution set for the inequality 3(8 – 4x) < 6(x – 5) is option C: A number line from negative 5 to 5 in increments of 1. An open circle is at negative 3, and a bold line starts at negative 3 and is pointing to the left.

To determine the solution set for the inequality 3(8 – 4x) < 6(x – 5), we need to solve it step by step:

Simplify the inequality:

24 - 12x < 6x - 30

Combine like terms:

-12x - 6x < -30 - 24

-18x < -54

Divide both sides of the inequality by -18, remembering to flip the inequality sign:

x > (-54) / (-18)

x > 3

The inequality tells us that x must be greater than 3. To represent this on a number line, we place an open circle at the value 3 and draw a bold line pointing to the right to indicate that the solution set includes all values greater than 3.

Therefore, option C accurately represent the solution set for the inequality 3(8 – 4x) < 6(x – 5).

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Answer:

[tex]m = \frac{1 - ( - 7)}{9 - ( - 3)} = \frac{8}{12} = \frac{2}{3} [/tex]

B is the correct answer.

The algebraic rule for translating the figure 3 units left and 2 units up is (x-3, y+2).  Option B.

To translate a figure 3 units to the left and 2 units up, we need to adjust the coordinates of the figure accordingly. The algebraic rule that represents this translation can be determined by examining the changes in the x and y coordinates.

When we move a figure to the left, we subtract a certain value from the x coordinates. In this case, we want to move the figure 3 units to the left, so we subtract 3 from the x coordinates.

Similarly, when we move a figure up, we add a certain value to the y coordinates. In this case, we want to move the figure 2 units up, so we add 2 to the y coordinates.

Taking these changes into account, we can conclude that the algebraic rule for translating the figure 3 units left and 2 units up is (x-3, y+2). The x coordinates are shifted by subtracting 3, and the y coordinates are shifted by adding 2. SO Option B is correct.

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Answer:

The correct answer is a.

Step-by-step explanation:

The sequence is: 2 3 E 4 5 I 6 8 We can notice that there are numbers and letters alternating in the sequence. The numbers are increasing, and the letters seem to be vowels in alphabetical order. So, the next value should be a letter (vowel) after I, which is O. The correct answer is a.

Answer:

There is one solution. The solution is 2, 18, 19.

Step-by-step explanation:

If you want me to show working tell me in the comments and I'll edit the answer

Answer:

A. (2, 18, -19)

Step-by-step explanation:

To solve:

Z is the most suitable variable to remove first

Add the first equation to the second equation: (this conveniently removes both y and z)

(x+y-z) + (4x-y+z) = 1+9

Simplify

5x = 10

Solve

x = 2

Multiply the second equation by 2 and minus it to the third equation: (Solve for y)

2(4x-y+z) - (x-3y+2z) = 2(9) - (-14)

Simplify

8x-2y+2z-x+3y-2z=18+14

7x+y=32

Substitute using x=2

7(2) + y = 32

y = 32 - 14

y = 18

Now substitute x and y for their respective values into Equation 1

2 + (-18) - z = 1

Simplify

-z = 19

z = -19

So :

x = 2, y = 18 , z = -19

The measure of the outside angle F indicated in the figure is 61 degrees,

The external angle theorem states that "the measure of an angle formed by two secant lines, two tangent lines, or a secant line and a tangent line from a point outside the circle is half the difference of the measures of the intercepted arcs.

Expressed as:

Outside angle = 1/2 × ( major arc - minor arc )

From the figure:

Major arc = 195 degrees

Minor arc = 73 degrees

Outside angle F = ?

Plug the value of the minor and major arc into the above formula and solve for the outside angle F:

Outside angle = 1/2 × ( major arc - minor arc )

Outside angle = 1/2 × ( 195 - 73 )

Outside angle = 1/2 × ( 122 )

Outside angle = 122/2

Outside angle = 61°

Therefore, the outside angle measures 61 degrees.

Option C) 61° is the correct answer.

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Answer:

8cm

Step-by-step explanation:

perimiter A = 11+11+4+4=30

perimiter B = 4+4+8+8=24

24+30=54

perimeter c =4+8+8+4+11+7+4=46

so perimiter of c is 8cm shorter than A and B total

hope this helps

Answer:

We have the slope of the line, which is 2 and a point that is (1, -1).

To find the point-slope form of the line, we use the equation:

y - y1 = m(x - x1)

where m is the slope and (x1, y1) is the given point.

Substituting in the values we have, we get:

y - (-1) = 2(x - 1)

Simplifying this equation, we get:

y + 1 = 2(x - 1)

Therefore, the answer is option C: y + 1 - 2(x - 1).

The tangent and normal lines of the curve:

Case (i): y = (4 / 5) · x + 13 / 5

Case (ii): (x, y) = (0, 13 / 5)

Case (iii): y = - (5 / 4) · x + 35 / 4

In this problem we have the representation of a curve whose equations for tangent and normal lines must be found. Lines are expressions of the form:

y = m · x + b

Where:

Both tangent and normal lines are perpendicular, the relationship between the slopes of the two perpendicular lines is:

m · m' = - 1

Where:

The slope of the tangent line is found by evaluating the first derivative of the curve at intersection point.

Case (i) - First, determine the slope of the tangent line:

y = √(8 · x + 1)

y' = 4 / √(8 · x + 1)

y' = 4 / √25

y' = 4 / 5

Second, determine the intercept of the tangent line:

b = y - m · x

b = 5 - (4 / 5) · 3

b = 5 - 12 / 5

b = 13 / 5

Third, write the equation of the tangent line:

y = (4 / 5) · x + 13 / 5

Case (ii) - Find the coordinates of the intercept of the tangent line:

(x, y) = (0, 13 / 5)

Case (iii) - First, find the slope of the normal line:

m' = - 1 / (4 / 5)

m' = - 5 / 4

Second, determine the intercept of the normal line:

b = y - m' · x

b = 5 - (- 5 / 4) · 3

b = 5 + 15 / 4

b = 35 / 4

Third, write the equation of the normal line:

y = - (5 / 4) · x + 35 / 4

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