Answers
Given:
• cotθ = -3
,• secθ < 0
,• 0 ≤ θ < 2π
Here the cot value of the angle is negative.
The cotangent function is negative in quadrants II and IV.
Also, secθ < 0, which means it is negative.
Secant function is negative in II and III quadrants.
Therefore, the angle will be in quadrant II.
Let's find the exact values of the following:
• (a). sin(2θ)
Apply the double angle formula:
[tex]sin(2\theta)=2sin\theta cos\theta=\frac{2tan\theta}{1+tan^2\theta}[/tex]Where:
[tex]tan\theta=\frac{1}{cot\theta}=-\frac{1}{3}[/tex]Thus, we have:
[tex]\begin{gathered} sin(2\theta)=\frac{2*(-\frac{1}{3})}{1+(-\frac{1}{3})^2} \\ \\ sin(2\theta)=\frac{-\frac{2}{3}}{1+\frac{1}{9}}=\frac{-\frac{2}{3}}{\frac{9+1}{9}}=\frac{-\frac{2}{3}}{\frac{10}{9}} \\ \\ sin(2\theta)=-\frac{2}{3}*\frac{9}{10} \\ \\ sin(2\theta)=-\frac{3}{5} \\ \\ \text{ Sine is positive in quadrant II:} \\ sin(2\theta)=\frac{3}{5} \end{gathered}[/tex]• cos(2θ):
Apply the formula:
[tex]cos(2\theta)=\frac{1-tan^2\theta}{1+tan^2\theta}[/tex]Thus, we have:
[tex]\begin{gathered} cos(2\theta)=\frac{1-(-\frac{1}{3})^2}{1+(-\frac{1}{3})^2} \\ \\ cos(2\theta)=\frac{1-\frac{1}{9}}{1+\frac{1}{9}} \\ \\ cos(2\theta)=\frac{\frac{9-1}{9}}{\frac{9+1}{9}}=\frac{\frac{8}{9}}{\frac{10}{9}}=\frac{8}{9}*\frac{9}{10}=\frac{4}{5} \\ \\ cos(2\theta)=\frac{4}{5} \\ \text{ } \\ \text{ Cosine is negative in quadrant II>} \\ cosine(2\theta)=-\frac{4}{5} \end{gathered}[/tex]• (c). sin(θ/2):
Apply the formula:
[tex]cos\theta=1-2sin^2(\frac{\theta}{2})[/tex]Where:
[tex]tan\theta=\frac{opposite}{adjacent}=-\frac{1}{3}[/tex]Now, let's find the hypotenuse using Pythagorean Theorem:
[tex]\sqrt{1^2+3^2}=\sqrt{1+9}=\sqrt{10}[/tex]Thus, we have:
[tex]cos\theta=\frac{adjacent}{hypotenuse}=-\frac{3}{\sqrt{10}}[/tex]Now, the function will be:
[tex]\begin{gathered} cos\theta=1-2sin^2(\frac{\theta}{2}) \\ \\ -\frac{3}{\sqrt{10}}=1-2sin^2(\frac{\theta}{2}) \\ \\ 2sin^2(\frac{\theta}{2})=1+\frac{3}{\sqrt{10}} \\ \\ 2sin^2(\frac{\theta}{2})=\frac{10+3\sqrt{10}}{10} \\ \\ sin^2(\frac{\theta}{2})=\frac{10+3\sqrt{10}}{20} \\ \\ sin(\frac{\theta}{2})=\sqrt{\frac{10+3\sqrt{10}}{20}} \end{gathered}[/tex]• (d). cos(,(θ/2)):
[tex]\begin{gathered} 2cos\theta=2cos^2(\frac{\theta}{2})-1 \\ \\ cos\frac{\theta}{2}=\sqrt{\frac{1+cos\theta}{2}}=\sqrt{\frac{1-\frac{3}{\sqrt{10}}}{2}} \end{gathered}[/tex]ANSWER:
[tex]\begin{gathered} (a).\text{ }\frac{3}{5} \\ \\ \\ (b).\text{ -}\frac{4}{5} \\ \\ \\ (c).\text{ }\sqrt{\frac{10+3\sqrt{10}}{20}} \\ \\ \\ (d).\text{ }\sqrt{\frac{1-\frac{3}{\sqrt{10}}}{2}} \end{gathered}[/tex]Related Questions
An archaeologist found a fossil that has a length of 459.89 in.Use the table of facts to find the length of the fossil in feet.Round your answer to the nearest tenth.Conversion facts for length12 inches (in) = 1 foot (ft)3 feet (ft) = 1 yard (yd)36 inches (in) = 1 yard (yd)5280 feet (ft) = 1 mile (mi)1760 yards (yd) = 1 mile (mi)-0 1ftХ??
Answers
Answer: 38.3ft
We need to convert 459.89 inches into feet. Given that 12 inches is equal to 1 foot, we can solve this by dividing the length by 12
[tex]459.89in\times\frac{1ft}{12in}[/tex]*Cancel the inches sign leaving us with ft
[tex]459.89\times\frac{1ft}{12}=\frac{459.89}{12}ft=38.32\approx38.3[/tex]Therefore 459.89 inches is equal to 38.3 feet.
Solve each equation by completing the square. X^2+10x=17
Answers
Completing squares
Before attempting to complete squares, let's recall the following identity
[tex](a+b)^2=a^2+2ab+b^2[/tex]the expression at the right side can be converted to the square of a binomial, provided we have the terms completed as shown
We have the equation:
[tex]x^2+10x=17[/tex]note the left side has TWO of the terms required for the square of a binomial. we only need the final number. but what number should we add?
the first term is the square of a, in this case, it's x
the second term has 10x and it should be 2ab, if we already know a=x, then
2ab=10x, then
b=10x/(2x)=5
now we know a=x and b=5, we only need to have b^2=25
that is exactly the number to add on both sides of the equation
[tex]x^2+10x+25=17+25=42[/tex]now we factor the left side:
[tex](x+5)^2=42[/tex]taking the square root, recall the square root can
An expression is shown. 14.1-(2.24*5); what is the value of the expression?
Answers
Given the expression:
[tex]14.1-(2.24\ast5)[/tex]Let's find the value of the expression.
To find the value of the expression, first evaluate the values in the parentheses:
[tex]\begin{gathered} 14.1-(2.24\ast5) \\ \\ =14.1-(11.2) \\ \\ \end{gathered}[/tex]Solving further:
[tex]\begin{gathered} 14.1-11.2 \\ \\ =2.9 \end{gathered}[/tex][tex]\begin{gathered} 14.1-(2.24\ast5) \\ \\ =14.1-(11.2) \\ \\ \end{gathered}[/tex]Solving further:
[tex]\begin{gathered} 14.1-11.2 \\ \\ =2.9 \end{gathered}[/tex][tex]undefined[/tex]Find the explicit formula for the geometric sequence. Then find a8. 4, 8, 16, 64, ...
Answers
The given sequence is : 4,8, 16,64
The geometric series is exoress as :
[tex]\text{ Geometric series=a, ar, ar}^2\ldots.ar^n[/tex]where r i the common ratio
In the given sequence the ratio is
[tex]\begin{gathered} r=\frac{\sec ond\text{ term}}{first\text{ term}} \\ r=\frac{8}{4} \\ r=2 \end{gathered}[/tex]So, the series will express as :
[tex]\begin{gathered} \text{Explict formula = 4(2)}^{n-1} \\ \text{Explict form = }4(2)^{n-1} \end{gathered}[/tex]Now for the 8 term
n=8
[tex]\begin{gathered} a_n=4\cdot2^{n-1} \\ a_8=4\cdot2^{8-1} \\ a_8=4\cdot2^7 \\ a_8=512 \end{gathered}[/tex]Answer : C) an=4.2^n-1, 512
Question 15 ptsIf Martha puts $181 in the bank today at 2%, how much will she have have in 8 years? (Round to the 2decimal places)Question 25 ptsHow much will Bill and Mary need to put in the bank today at 5% to have $103,897 in 9 years? (Roundto 2 decimal places)
Answers
Question 1
Interest in 1 year at 2% per annum = 2/100 x $181 = $3.62
Total Interst after 8 years = 8 x $3.62 = $28.96
Total amount she has after 8 years = initial amount + Total Interst after 8 years
=$181 + $28.96 = $209.96 (2 decimal places)
Question 2
rate per annum, r = 5%
Time = 9years
Total amount = $103,897
Let the initial amount invested be p
interst after 1 year = 5% of p =$5p/100
Total interest after 9 years = 9 x 5p/100 = $45p/100
Total amount = p + 45p/100
That is,
103,897= p + 45p/100
[tex]undefined[/tex]A board game uses a spinner like one below, where 0, 1, 2, and 3 are all equally likely.Each turn, a player spins twice and subtracts the results of the spins. The game only looks at non-negativedifferences. For example, if a player spins a 1 and a 3, the difference is 2.Let X represent the difference in given turn.Which tables represents the theoretical probability distribution of X?Choose 1 answer:
Answers
The possible outcomes are:
0: 1-1, 2-2, 3-3, 4-4 => 4
1: |2-1|, |3-2|, |4-3|, |1-2|, |2-3|, |3-4| => 6
2: |3-1|, |4-2|, |1-3|, |2-4| => 4
3: |4-1|, |1-4| => 2
The total number of outcomes is 4+6+4+2 = 16. Then, the table that represents the theoretical distribution is:
Answer:
It should be B, the one that has 0,1,2,3 on the top.
Step-by-step explanation:
right on khan
What is the image of (8,−4) after a dilation by a scale factor of 1/4 centered at the origin?
Answers
Answer:
[tex](2, -1)[/tex]
Step-by-step explanation:
When dilating with a scale factor of [tex]k[/tex] about the origin, [tex](x,y) \longrightarrow (kx, ky)[/tex].
Write 2.78 x 10-4in standard form.
Answers
Given ,
The scientific notation of the equation is,
[tex]2.78\times10^{-4}[/tex]The standard notation of the scientific notation is,
[tex]\begin{gathered} 2.78\times\frac{1}{10^4} \\ =\frac{2.78}{10000} \\ =0.000278 \end{gathered}[/tex]Hence, the standard form is 0.000278.
AGE VS. STATES TRAVELED TO Age States (years) Traveled to 54 46 Linear Regression: y=0.111x+7.668 slope: 12 5 3 1 Y-intercept: 25 17 35 2 Use your equation to predict the age of a person if he/she travelled to 20 states. 68 4 104 12
Answers
The linear regression equation is expressed as
y = 0.
Question 5 (1 point)A student takes a multiple-choice test with 8 questions on it, each of which have 4 choices. The student randomlyguesses an answer to each question.What is the probability that the student gets exactly 4 questions correct?Round to 3 decimal places.a0.886b0.9730.0870.208d
Answers
We can use Binomial distribution to calculate the probability of exactly 4 success
There are 8 questions which is our trial
Probability of succes (p)
Since in every question, there is 4 options with one right answer, then probability of success (p) = 1/4 = 0.25
probabiliti of failure (q) = 1- p = 1- 0.25 = 0.75
We will now use the formula below
[tex]p(x)^{}=^nC_xP^xq^{n-x}[/tex]substitute the values into the formula
[tex]p(x=4)=^{8\text{ }}C_4(0.25)^4(0.75)^{8-4}[/tex][tex]=\frac{8!}{(8-4)!4!}.(0.25)^4.(0.75)^4[/tex][tex]=\frac{8!}{4!4!}\text{.}(0.25)^4(0.75)^4[/tex][tex]=\frac{8\times7\times6\times5\times4!}{4\times3\times2\times1\times4!}\times(0.25)^4\times(0.75)^4[/tex][tex]=\frac{1680}{24}\times(0.00390625)\times(0.31640625)[/tex][tex]\approx0.087[/tex]13/50 make as a decimal
Answers
To convert the fraction 13/50 to decimal, we have to divide 13 by 50.
We can also calculate it as:
[tex]\frac{13}{50}=\frac{13}{50}\cdot\frac{2}{2}=\frac{26}{100}=0.26[/tex]In this case, we multiply both the numerator and the denominator by 2. Then, we obtain a denominator of 100, which means that the numerator is in hundredths. Then, we can write is as 0.26.
Answer: the decimal expression of 13/50 is 0.26.
A _____ is a line that best approximates the linearrelationship between two variables in a data set. *
Answers
We have the following:
A line of best fit is a straight line that is the best approximation of the given data set. It is used to study the nature of the relationship between two variables.
Therefore, the answer is line of best fit.
A line of best fit is a line that best approximates the linear
relationship between two variables in a data set.
According to the data in the table which country has a more population density ?
Answers
The first step to solve the question presented is to calculate the population density for each country, which is defined as the ratio from the population and the area of the country, as follows:
[tex]D=\frac{P}{A}[/tex]Let us to perform the calculation for both countries with data in the table.
[tex]\begin{gathered} D_{\text{America}}=\frac{310,000,000}{3,539,225}\cong87.59\frac{people}{mi^2} \\ D_{\text{Mexico}}=\frac{122,000,000}{742,485}\cong164.31\frac{people}{mi^2}_{} \end{gathered}[/tex]From the solution presented we are able to conclude that the country with the highest population density from those in the table is Mexico
Solve and graph the following inequality. 6c-12>42
Answers
We have the next inequality
[tex]6c-12>42[/tex]And we must solve and graph it.
First, we need to solve the inequality
To solve the inequality we must:
1. Add 12 to both sides
[tex]\begin{gathered} 6c-12+12>42+12 \\ \text{ Simplifying,} \\ 6c>54 \end{gathered}[/tex]2. Divide both sides by 6
[tex]\begin{gathered} \frac{6c}{6}>\frac{54}{6} \\ \text{ Simplifying,} \\ c>9 \end{gathered}[/tex]So, the solution of the inequality is c > 9
Finally, we must graph it
We can see that the solution are all values greater than 9, so the graph would be
that the triangles below are congruent select all options that would provide enough information to prove the triangles are congruent
Answers
The correct options are options C and D;
1) Angle C is congruent to E
2) BA and FD are congruent
Here, we want to select the options that would prove that the triangles are congruent
We can get that from the markings on the diagram
As we can see from the question, we already have two sides being equal as we can see from the markings
Hence, we need an extra information to see through that the two triangles are indeed congruent
If the sides BA and FD are equal, it will simply mean that the three sides of the triangle are equal. In that case, the two triangles are congruent by SSS (side-side-side)
Also, if the angles C and E are congruent, we will have that the two triangles are similar by the SAS (side-angle-side)
So, the correct options are C and D
"A pump can fill a tank in 3 hours. A more powerful pump can fill the same tank in 2 hours. How long would it take to fill the tank with both pumps working?" Is my answer correct or incorrect? 1/3+1/2=3+2/3*2 5/15=15/515/5=3hrs
Answers
A pump can fill a tank in 3 hours. A more powerful pump can fill the same tank in 2 hours. How long would it take to fill the tank with both pumps working?" Is my answer correct or incorrect?
we have that
fisrt pump ------> fill a tank in 3 hours
that means
100% -----> 3 hours
1 hour ------> 33.33%
second pump
fill the same tank in 2 hours
100% -----> 2 hours
1 hour -----> 50 %
therefore
with both pumps working
1 hour ------> (33.33%+50%)=83.33%
applying proportion
1/83.33=x/100
x=100/83.33
x=1.2 hoursyour answer is not correctbecause the total time must be less than 2 hours (time of the second pump)The amount of metal needed to be installed spring the workbench is
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To answer this question we will use the following formula to compute the perimeter of a rectangle:
[tex]Perimeter=2(length+width)[/tex]Therefore the perimeter of the workbench is:
[tex]Perimeter=2(5ft+3ft).[/tex]Simplfying the above result we get:
[tex]Perimeter=2(8ft)=16ft.[/tex]Therefore we will need 16ft of metal stripping.
Answer: 16ft.
PLEASE HELP ME!!!!!
Subtract the linear expressions.
(-3 + 4x - 9x) - (9 - 11x + 7)
Answers
6x + 19 is the difference between both the linear expression.
What is linear equation ?In a linear equation, the variable's greatest power always equals 1. It is also referred to as a one-degree equation. The usual form of a linear equation with one variable is written as Ax + B = 0. In this case, x is a variable, A is a coefficient, and B is a constant.
Calculation-3 + 4x -9x - 9 + 11x - 7
= -19+6x
= 6x + 19
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Answer:
Between the two linear expressions, there is a difference of 6x + 19.
How do linear equations work?
In a linear model, the largest power of the variable is always equal to 1. Another name for it is a one-degree equation. Ax + B = 0 is how a statistical model with one term is typically written. In this scenario, the variables x and A are coefficients, while B is a standard.
Calculation -3+4x-9x-09+11x-7-9
= -19+6x
= 6x + 19
Step-by-step explanation:
I'm also in k12 hope this helps! :3
A Statistics exam has mean m=78 in standard deviation of g=8 estimate the portion of Grades between 66 and 90
Answers
In order to estimate the portion of grades between 66 and 90, first let's find the z-score for these two values, using the formula:
[tex]z=\frac{x-m}{g}[/tex]So we have:
[tex]\begin{gathered} z_1=\frac{66-78}{8}=-1.5 \\ z_2_{}=\frac{90-78}{8}=1.5 \end{gathered}[/tex]Looking at the z-table, a z-score of 1.5 corresponds to a z-value of 0.0668.
Since we have this z-value from the left and right, the percentage we want is:
[tex]\begin{gathered} P=1-0.0668-0.0668 \\ P=0.8664=86.64\text{\%} \end{gathered}[/tex]The probability distribution of a random variable x is given in the table below.X10-505101520Probability.2015.05.1.25.1.15Find the probability that x ⩾ 5
Answers
Answer:
0.6
Explanation:
From the given data:
[tex]\begin{gathered} P\left(x=-10\right)=0.20 \\ P\left(x=-5\right)=0.15 \\ P\left(x=0\right)=0.05 \\ P\left(x=5\right)=0.1 \\ P\left(x=10\right)=0.25 \\ P\left(x=15\right)=0.1 \\ P\left(x=20\right)=0.15 \end{gathered}[/tex]The probability that x is greater than or equal to 5 is:
[tex]\begin{gathered} P(x\geq5)=P\left(x=5\right)+P\left(x=10\right)+P\left(x=15\right)+P\left(x=20\right) \\ =0.1+0.25+0.1+0.15 \\ =0.6 \end{gathered}[/tex]The proportion of passengers who miss a flight for which they have a reservation is0.0995. Suppose a flight 290 reservations. Find the standard deviation of the sampleproportion, ºf, rounded to the nearest ten-thousandth (4 decimal places).
Answers
Answer
Standard deviation of the sample proportion = 0.0176
Explanation
For a distribution with proportion, p, the standard deviation of the sample proportion is given as
[tex]\sigma_x=\sqrt[]{\frac{p(1-p)}{n}}[/tex]where
p = sample proportion = 0.0995
n = sample size = 290
[tex]\begin{gathered} \sigma_x=\sqrt[]{\frac{p(1-p)}{n}} \\ \sigma_x=\sqrt[]{\frac{0.0995(1-0.0995)}{290}} \\ \sigma_x=\sqrt[]{\frac{0.0995(0.9005)}{290}} \\ \sigma_x=\sqrt[]{\frac{0.08959975}{290}} \\ \sigma_x=\sqrt[]{0.0003089647} \\ \sigma_x=0.0176 \end{gathered}[/tex]Hope this Helps!!!
Substitution, SUVAT
Use the correct equation below to work
out the terms required:
v=u + at, or v² = u² + 2as
a) u = 6ms ¹, a = 3ms2, v = 30ms ¹, find t.
b) a = 0.5ms², v = 10.5ms¹, t = 5s, find u.
c) u = 2ms ¹, v = 6ms¹, a = 0.5ms², find s.
Answers
The value of time (t) is equal to 8 seconds.
The value of the initial velocity (u) is equal to 8 meters per seconds.
The value of the distance (s) is equal to 32 meters.
How to find the missing terms?Mathematically, the first and third equation of motion are given by this mathematical expressions:
v = u + at
v² = u² + 2as
Where:
V represents the final velocity.U represents the initial velocity.S represents the distance travelled or covered.t represents the time measured in seconds.For the first part (a), we would find time (t) by using the first equation of motion as follows;
v = u + at
30 = 6 + 3t
3t = 30 - 6
3t = 24
Time, t = 24/3
Time, t = 8 seconds.
For the second part (b), we would find the initial velocity (u) by using the first equation of motion as follows;
v = u + at
10.5 = u + 0.5(5)
10.5 = u + 2.5
Initial velocity (u) = 10.5 - 2.5
Initial velocity (u) = 8 m/s.
For the third part (b), we would find the distance (s) by using the third equation of motion as follows;
v² = u² + 2as
6² = 2² + 2(0.5)s
36 = 4 + s
Distance, s = 36 - 4
Distance, s = 32 meters.
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In the diagram from question 15, which statement would prove that line a and b are parallel?
Answers
the answer is D
[tex]m\measuredangle1+m\measuredangle7=180[/tex]2. Zero can be a negative number.OTrueFalse
Answers
In the real numbers system, any negative number x meets the following property:
[tex]x<0[/tex]The symbol "<" means that x is smaller than 0 but never equal to 0 so the definition of negative numbers excludes the zero. Then this statement is False.
What is the quotient and the remainder of 26÷3
Answers
Answer:
Quotient: 8
Remainder: 2
Explanation:
If we divide 26 by 3, we get:
So, the quotient is 8 and the remainder is 2.
Answer: Quotient: 8 Remainder: 2
Step-by-step explanation: 26/3 = 8 r2
1) Write an equation of the line perpendicular to: y = 3x - 9 with a y-intercept of 4.
Answers
The given equation is
[tex]y=3x-9[/tex]The new line is perpendicular to the given equation, which means we have to use the following formula.
[tex]m\cdot m_1=-1[/tex]Where the slope of the given line is 3 (the coefficient of x).
[tex]m\cdot3=-1[/tex]We solve for m.
[tex]m=-\frac{1}{3}[/tex]So, the slope of the new perpendicular line is -1/3.
According to the problem, the y-intercept of the new perpendicular line is 4. Now, we use the slope-intercept form to write the equation.
[tex]\begin{gathered} y=mx+b \\ y=-\frac{1}{3}x+4 \end{gathered}[/tex]Therefore, the equation of the new line is[tex]y=-\frac{1}{3}x+4[/tex]How much will a customer spend on a sweater that is $65.00 but discounted 20% and purchased in a state that has an 8% sales tax?
Answers
1) Gathering the data
Sweater $65
Discounted 20%
Tax: 8%
2) We can find this final price using this formula/calculation
Since the price has been discounted by 20% we can multiply $65 x 0.8 to find the discounted price, but the sweater will be sold with a tax of 8% so we can multiply by (1 + 0. 08) to get the final price, i.e. $56.16
A
(6,3) and (2, -9)equation in slope intercept form
Answers
Let:
(x1,y1)=(6,3)
(x2,y2)=(2,-9)
[tex]\text{slope}=m=\frac{y2-y1}{x2-x1}=\frac{-9-3}{2-6}=\frac{-12}{-4}=3[/tex]Using the point-slope equation:
[tex]\begin{gathered} y-y1=m(x-x1) \\ y-3=3(x-6) \\ \text{Solve for y:} \\ y=3x-15 \end{gathered}[/tex]The value of tan(alpha+beta) given sin(alpha)=40/41 and sin(beta)=15/17 and cos(∝ + β)= -528/697 and sin(∝ + β)= 455/679
Answers
Answer
Tan (∝ + β) = -455/528
Explanation
Tan (∝ + β) = ?
Given:
Sin ∝ = 40/41
Sin β = 15/17
Cos (∝ + β) = -528/697
Sin (∝ + β) = 455/697
Note: Tan ∝ = Sin ∝/Cos ∝
⇒ Tan (∝ + β) = (Sin ∝ + β)/(Cos ∝ + β)
Recall Sin (∝ + β) = 455/697 and Cos (∝ + β) = -528/697
∴ Tan (∝ + β) = (455/697)/(-528/697)
Tan (∝ + β) = 455 /697 x (-697/528)
Tan (∝ + β) = -455/528
The stock market opens on Monday morning, a stock was valued at 42.50. The value of that stock increase by the same amount each day for the next three days. After three days, the value of stock was 50.00. Which equation is used to find x, the amount the stock rose in value each day.
Answers
The stock market opens on Monday morning, a stock was valued at 42.50.
The intial value of stock = 42.50
The value of that stock increase by the same amount each day for the next three days.
Let the increase amount in each day is x
then, increase amount in three day = 3x
After three days, the value of stock was 50.00.
Net amount after three day = 50.00
Net amount after three day = Initial value of stock + Incease amount in three day
50 = 42.50 + 3x
3x + 42.50 = 50
Answer: 3x + 42.50 = 50