Okey, here we have the following:
Capital: 10000
Interest: 2%
Time: 7 Years= 7*4=28 quarters of year
Using the compound interest formula, we get:
[tex]C_f=10000(1+\frac{0.02}{4})^{4\cdot7}=1000(1+\frac{0.02}{4})^{28}[/tex]Working we get:
[tex]C_f\approx11.498.73[/tex]She will have aproximately $11,498.73 in her account after 7 years.
162-317-3113-510Is this relation a function?
can you see my messages?
I don’t understand how to get the second x intercept
In this problem
the vertex is given ------> (40/2,12)-------> (20,12)
The first intercept is (0,0)
therefore
second intercept is
x-intercept=20+20=40
(40,0) is the coordinates of the second x-intercept
(the vertex is the midpoint between the first and second x-intercept)
see the attached figure
2. Given: ZMOP is a right angle RP I OP Prove: MO || RP
Given that;
[tex]\begin{gathered} \measuredangle MOP\text{ is a right angle.} \\ \measuredangle MOP=90^0 \end{gathered}[/tex]And;
[tex]\vec{RP}\perp\vec{OP}[/tex]Since line RP is perpendicular to line OP, Angle RPO must be a right angle.
[tex]\measuredangle RPO=90^0[/tex]Recall that for two parallel lines intersected by a straight line, Same side interior angles are supplementary.
[tex]A+B=180^0[/tex]So, for line MO to be parallel to line RP, the sum of angle MOP and angle RPO must be equal to 180 degree.
[tex]\measuredangle MOP+\measuredangle RPO=90+90=180^0[/tex]Since the sum of angle MOP and angle RPO is equal to 180 degree, then line MO is parallel to line RP.
[tex]\begin{gathered} \text{ Since} \\ \measuredangle MOP+\measuredangle RPO=180^0 \\ \text{Then;} \\ MO\Vert RP \end{gathered}[/tex]Proved
WhaGraph the piecewise-defined function. Use the graph to determine the domain and range of the function. x + 2 if x < -1F(x)={ - 2x + 3 if x ≥ - 1
The domain of the function is all possible x-values a function can have; therefore, we see here that the domain of the function is all real numbers (including -1).
The range of a function is all possible y values a function can take. We see from the graph above that can take only the values that are greater than or equal to 1; therefore, the range of the function is all real numbers greater than or equal to 1.
It takes 6 eggs, 5 oz of cheese, and 2 oz of butter to make twoomelets. What is the cost per omelet if eggs cost $.99 per dozen,1 lb of cheese costs $4.29, and 1/2 lb of butter costs $1.25?a. $2.15b. $1.34c. $1.08d. $.31
Given:
It takes 6 eggs, 5 oz of cheese, and 2 oz of butter to make two
omelets
Eggs cost per dozen = $0.99
So, the cost of 6 eggs = 0.99/2 = 0.495
1 lb of cheese costs $4.29
1 lb = 16 oz
So, the cost of 5 oz =
[tex]\frac{5}{16}\cdot4.29=1.34[/tex]1/2 lb of butter costs $1.25
So, the cost of 2 oz =
[tex]\frac{2}{8}\cdot1.25=0.3125[/tex]So, the cost of two omelets = 0.495+1.34+0.3125 = 2.1475
So, the cost of one omelet = 2.1475/2 ≈ 1.08
So, the answer will be option c. $1.08
solve each system by substitution.y =-2x + 5y =-8x+17
To solve the equation system by substitution, since the equations are expressed in terms of y, you have to equal both expressions and calculate the value of x:
[tex]\begin{cases}y=-2x+5 \\ y=-8x+17\end{cases}[/tex][tex]\begin{gathered} y=y \\ -2x+5=-8x+17 \end{gathered}[/tex]To calculate the value of x, the first step is to pass the x-term to the left side of the equation by applying the opposite operation:
[tex]\begin{gathered} -2x+8x+5=-8x+8x+17 \\ 6x+5=17 \end{gathered}[/tex]Next, pass 5 to the right side of the equation:
[tex]\begin{gathered} 6x+5-5=17-5 \\ 6x=12 \end{gathered}[/tex]Finally, divide both sides by 6 to reach the value of x
[tex]\begin{gathered} \frac{6x}{6}=\frac{12}{6} \\ x=2 \end{gathered}[/tex]Now that we have determined the value of x, replace it in either one of the original equations to determine the value of y:
[tex]\begin{gathered} y=-2x+5 \\ y=-2\cdot2+5 \\ y=-4+5 \\ y=1 \end{gathered}[/tex]The solution for this equation system is (2,1)
Determine if the 2 lines are parallel, perpendicular, or neither based on their slope- intercept equations.
Equations of lines H & I;
Line H: y=z
Line I: y=-7z - 33
O Not Enough Information
O Perpendicular
O Neither
POSSIBLE PO
O Parallel
Equations of lines H & I; Line H: y=z Line I: y=-7z - 33 is Perpendicular. The lines are not parallel if the slopes differ. Perpendicular lines do meet, but parallel lines do not.
How can you demonstrate that two lines in an equation are parallel?Only if the slopes of two lines are equal can they be said to be parallel. The conventional version of the equation is 2x - 3y = 4. Since a line with the equation Ax + By = C typically has a slope of -A/B, line q must have a slope of -2/-3 = 2/3.
Their equations allow us to compare the slopes of two lines to determine if they are parallel. The lines are parallel if the slopes are the same and the y-intercepts are different.
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A flagpole casts a shadow 3.5 meters long, Anita is standing near the pole. Her shadow is 0.75 meters long, Anita's height is 1.5 meters.How tall is the flagpole? Draw a diagram, label, and solve. Type your answer as a whole number or a decimal with no labels. EX5.2
Solution:
The heights and the shadows are in the same ratio because the sun is shining from the same angle, so the triangles formed are similar.
Notice that Anita's height is twice as long as her shadow, so the height of the flagpole will be
[tex]2\text{ x }3.5\text{ = 7m}[/tex]We can also write a direct proportion:
[tex]\frac{x}{3.5}\text{ = }\frac{1.5}{0.75}\text{ }\frac{\leftarrow\text{heights}}{\leftarrow shadows}[/tex]solving for x, we get:
[tex]x\text{ =}\frac{3.5\text{ x }1.5}{0.75}\text{ = 7m}[/tex]then, we can conclude that the correct answer is:
[tex]x\text{ = 7m}[/tex]Samantha received a loan from the bank for $4,500. She plans on payinyoff the loan in 4 years. At the end of 4 years, Samantha will have paid$900 in interest. What is the simple interest rate on the bank loan?
The simple interest rate formular is;
I = A - P
A= I + P
A = P ( 1 + rt )
A is the amount after t years
P is the initial amount = $4,500
r is the rate in percent = ?
t is the time in years = 4
A = $4,500 + $900 = $5,400
Therefore to obtain the rate (r)
5400 = 4500 (1 + r x 4 )
1 + 4r = 5400/4500
1 + 4r = 1.2
4r = 1.2 - 1
4r = 0.2
r = 0.2/4 = 0.05
In percentage;
r = 0.05 x 100 = 5%
Thus, the simple interest rate is 5%
Assume the normal distribution of data has a mean of 14 and a standard Deviation of 3. use the 65-95-99.7 rule to find the percentage of values that lie below 8
By the 65-95-99.7 rule,
[tex]\begin{gathered} 65\text{ \% of the distribution lies below }\bar{x}+\sigma\text{ and above }\bar{x}-\sigma \\ 95\text{ \% of the distribution lies below }\bar{x}+2\sigma\text{ and above }\bar{x}-2\sigma \\ 99.7\text{ \% of the distribution lies below }\bar{x}+3\sigma\text{ and above }\bar{x}-3\sigma \end{gathered}[/tex]By symmetry,
[tex]\begin{gathered} 47.5\text{ \% of the distribution lies above }\bar{x}-\sigma\text{ and below }\bar{x} \\ \text{ Hence,} \\ 2.5\text{ \% of the values lies below }\bar{x}-\sigma \end{gathered}[/tex]In our case,
[tex]\bar{x}=14,\sigma=3[/tex]Therefore,
[tex]\begin{gathered} 8=14-6=14-2(3) \\ \text{Hence,'} \\ 8=\bar{x}-2\sigma \end{gathered}[/tex]Hence, 2.5 % of the values lie below 8
Mark went to the bank to borrow $10,000. He was given 2 options for a $10,000 loan:OPTION 1: 24-month payback at 6%interest will result in a monthly payment of $443.21 per month, or OPTION 2: 36-month payback at 6% interest will result in a monthly payment .of $304.22 per month.Which statement is NOT true?F. Mark will pay a total of $10,637.04 if he chooses Option 1.G. Mark will pay a total of $10,951.92 if hechooses Option 2.H. Mark will save $314.88 if he selects Option 2.J. Mark will pay a lower total amount if he selects Option 1.
Loan= $10.000
Bank options:
24-month payback 6% interest, with a monthly payment of $443.21/month
Then, Mark in option 1 will pay a total of:
[tex]443.21\text{ x 24 months=}10,637.04\text{ in option 1. }[/tex]36-month payback 6% interest, with a monthly payment of $304.22/month.
Mark in option 2 will pay a total of:
[tex]304.22\text{ x 36months=}10,951.92\text{ in option 2. }[/tex]Then, Mark will pay a lower total amount of money if he selects option 1 (10.637.04 is less than 10,951.92), saving a total of:
[tex]10,951.92\text{ - 10.637.04= 314.88 if he chooses option 1. }[/tex]Therefore, the statement that is NOT true is:
H. Mark will save $314.88 if he selects option 2.
A kite is flying 95 ft off the ground, and its string is pulled taut. The angle of elevation of the kite is 48º. Find the length ofthe string. Round your answer to the nearest tenth.
Given data:
Kite is flying off the ground = 95ft. ( Perpendicular)
Angle = 48 degree
[tex]\sin 48^{\circ}=\frac{Perpendicular}{Hypotenues}[/tex][tex]\text{Hypotenues}=\frac{Perpendicular}{\sin 48^{\circ}}[/tex][tex]\begin{gathered} H=\frac{95}{0.7431} \\ H=127.84ft \end{gathered}[/tex]Thus, the length of the string is 127.8 ft.
Amy’s grandmother is exactly 6 times older than Amy.
Which three statements below must be true?
The three statements that are true include:
Amy's age is a factor of the age of her grandmother.
The age of Amy's grandmother is a composite number.
The number 6 is a factor if the age of Amy's grandmother.
What is a factor?A factor simply means a number that can be multiplied by another number to get the original number.
Let's say Amy is 10 years. The grandmother will be 69 years. In this case, 10 is a factor of 60. Therefore, Amy's age is a factor of the age of her grandmother.
The complete options are given below.
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Which three statements below must be true?
1.The age of Amy’s grandmother is a prime number.
2.Amy’s age is a factor of the age of her grandmother.
3.The age of Amy’s grandmother has exactly two factors.
4.The age of Amy’s grandmother is a composite number.
5.The number 6 is a factor of the age of Amy’s grandmother.
6.The age of Amy’s grandmother has exactly four factors.
The administrator at your local hospital states that on weekends the average wait time for emergency room visits is 11 minutes. Based on discussions you have had with friends who have complained about how long they wait to be seen in the ER over a weekend, you dispute the administrator's claim. You decide to test your hypothesis. Over the course of a few weekends, you record the wait time for 28 randomly selected patients. The average wait time for these selected patients is 12 minutes with a standard deviation of 2.5 minutes. Do you have enough evidence to support your hypothesis that the average ER wait time is longer than 11 minutes? Conduct your test with a 5% level of significance.
This is a hypothesis test for the population mean.
The claim is that on weekends the average wait time for emergency room visits is more than 11 minutes.
Then, the null and alternative hypothesis are:
[tex]\begin{gathered} H_0\colon\mu=11 \\ H_a\colon\mu>11 \end{gathered}[/tex]The significance level is 0.05.
The sample has a size n=28.
The sample mean is M=12.
As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=2.5.
The estimated standard error of the mean is computed using the formula:
[tex]s_M=\dfrac{s}{\sqrt{n}}=\dfrac{2.5}{\sqrt{28}}=0.4725[/tex]Then, we can calculate the t-statistic as:
[tex]t=\dfrac{M-\mu}{s/\sqrt{n}}=\dfrac{12-11}{0.4725}=\dfrac{1}{0.4725}=2.117[/tex]The degrees of freedom for this sample size are:
[tex]df=n-1=28-1=27[/tex]This test is a right-tailed test, with 27 degrees of freedom and t=2.117, so the P-value for this test is calculated as (using a t-table):
[tex]\text{P-value}=P(t>2.117)=0.0218[/tex]As the P-value (0.0218) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
Conclusion: at a significance level of 0.05, there is enough evidence to support the claim that, on weekends, the average wait time for emergency room visits is more than 11 minutes.
find the surface area of the cone in terms of pi. SA=__ cm squared. simply
Given the figure of a cone.
As shown, the slant height = s = 23 cm
And the diameter of the base = d = 18 cm
So, the radius = r = 0.5d = 9 cm
The surface area of the cone will be calculated using the following formula:
[tex]SA=\pi rs+\pi r^2[/tex]Substitute s = 23, and r = 9, writing the surface area in terms of π
[tex]SA=π(18)(23)+π(9)^2=414π+81π=495π[/tex]So, the answer will be:
The surface area of the cone = 495π cm²
The following probability table shows probabilities concerning Favorite Subject and Gender. What is the probability of selecting an individual who is a female or prefers science?
Gender Favorite Subject Total
Math English Science
Male 0.200 0.050 0.175 0.425
Female 0.100 0.325 0.150 0.575
Total 0.300 0.375 0.325 1.000
Answer: 2
Step-by-step explanation: 0.300 0.375 0.325 1.000 = 2
Janet has a scale drawing in her room
Using the scale drawing, we can see that dimensions of the room are 17 feet by 14 feet.
So the correct option is A.
What are the actual dimensions of the room?
We know that each inch in the scale drawing is equal to 5 feet in the real room, in this case we know that the length of the drawing is 3.4 inches, then the real length is:
L = 3.4*5 ft = 17ft
And the width in the drawing is 2.8 inches, then the real width is:
W = 2.8*5 ft = 14ft
Then the dimensions of the room are 17 feet by 14 feet.
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309+23143240-59234881
Given data:
The given numbers are 309+23143240-59234881.
The simplification of the given numbers is,
-36091332.
nowledge Check 01
On November 1, the company rented space to another tenant. A check in the amount of $9,000, representing three months' rent in advance, was received from the tenant on that date. The payment was recorded with a credit to the Unearned Rent Revenue account.
Complete the necessary December 31 adjusting journal entry by selecting the account names from the pull-down menus and entering dollar amounts in the debit and credit columns.
Debit for unearned rent revenue of $6,000.
Rent Revenue Credit $6,000
What is known as the revenue?The total amount of revenue produced by the purchase of goods or services linked to the company's main operations is referred to as revenue. Because it appears at the top of the revenue statement, revenue, also termed as total sales, is often made reference to as the "top line."For the given question,
It is assumed that the company rented area to another tenant on November 1. On that date, the tenant handed over a check for $9,000, which represented three months' rent in advance. The payment has been recorded as a credit to the account for unearned rent revenue.Now, on December 31, we must prepare this same adjusting entry to record this same Rent Revenue for the two-month period (Nov. 1 to Dec. 31).
For two months, the rent revenue will be 9000×2/3 = $6,000
As a result, the journal entry to track the Rent revenue is as follows:
Debit for unearned rent revenue of $6,000.
Rent Revenue Credit $6,000
(becoming the improvement made for earned Rent Revenue).
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A translation 6 units right maps P onto P'. Complete the translation function.
If we have a point P=(x,y) and we apply a translation 6 units to the right we will get a point P' that is:
[tex](x,y)\longrightarrow(x+6,y)[/tex]We can test it by trying with P=(0,0).
Then P' would be (6,0), that is 6 units to the right from P.
Answer: (x,y) --> (x+6,y)
random variables, probability distributions and expected value Alyssa likes to play roulette, but she doesn't like the low probability of betting on a single number. Therefore, she bets on a block of 4 numbers, increasing her probability of winning to 38. She generally places a $5 chip on her block of 4. If any other number comes up she loses her bet, but if one of her 4 numbers come up, she wins $40 (and gets to keep her bet!). What is the expected value for Alyssa playing roulette? Round to the nearest cent. Do not round until your final calculation.
We have to calculate the expected value for Alyssa playing roulette.
The expected value is calculated as the weighted sum of all the possible the outcomes, weighted by the probabilities of occurrence of this outcomes.
Then, we start by listing all the outcomes:
1) One of the numbers of the block comes up.
This will happen with a probability of 4 out of 38 (P=4/38). NOTE: The total numbers of the roulette are 38.
The net prize, that is excluding the $5 she bets, is $40.
2) None of the numbers of the block comes up.
That will happen with probability 34 out of 38 (P=34/38).
The net prize, as she will lose the $5 she bets, is -$5.
The expected value can be calculated as:
[tex]E=\sum ^2_{i=1}p_i\cdot X_i=\frac{4}{38}\cdot40+\frac{34}{38}\cdot(-5)=\frac{160}{38}-\frac{170}{38}=\frac{-10}{38}\approx-0.26[/tex]The expected value for Alyssa is -$0.26.
Calculate the variance and the standard deviation for the following set of data: 7, 2, 5, 3, 3, 10
We need to know about variance and standard deviation to solve the problem. The variance of the set is 7.67 and the standard deviation is 2.77
Variance is a measure of dispersion which means it measures how far a set of numbers is spread out from the mean value. Standard deviation is the square root of variance. Inorder to calculate the variance we need to calculate the mean of the data set first.
mean=7+2+5+3+3+10/6=30/6=5
variance=[(7-5)^2+(2-5)^2+(5-5)^2+2(3-5)^2+(10-5)^2]/6=4+9+8+25/6=46/6=7.67
standard deviation =[tex]\sqrt{var}[/tex]=2.77
Therefore the variance of the data set is 7.67 and the standard deviation is 2.77
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Simplify. 3 6 4 2m n 4 6m Write your answer using only positive exponents. . X Х ?
we have the expression
[tex](\frac{2m^6n^4}{6m^4})^3[/tex][tex](\frac{2m^6n^4}{6m^4})^3=\frac{(2^3)(m^{(18)})(n^{(12)})}{(6^3)(m^{(12)})}[/tex]simplify
[tex]\frac{(8)(m^{(18-12)})(n^{(12)})}{216}=\frac{(m^6)(n^{(12)})}{27}[/tex]Tents-R-Us makes and sells tents. Tents-R-Us' motto is“Keep It Simple.” The company decides to makes justthree sizes of tents: the Mini, the Twin, and theFamily-Size. All the tents they make have equilateraltriangular ends as shown at right.1. For the Twin, each edge of the triangle will be 8 ft. Find the heightof the tent at the center, correct to the nearest inch. One way to findthis height is to make an accurate scale drawing and measure.
The company decides to make just three sizes of tents: the Mini, the Twin, and the Family-Size.
The shape of these tents is an equilateral triangle.
Part 1:
For the Twin, each edge of the triangle will be 8 ft.
The height of the tent is given by
[tex]h=a\cdot\frac{\sqrt[]{3}}{2}[/tex]Where a is the length of the edge of the triangle.
Since we are given that a = 8 ft
[tex]\begin{gathered} h=a\cdot\frac{\sqrt[]{3}}{2} \\ h=8\cdot\frac{\sqrt[]{3}}{2} \\ h=4\sqrt[]{3} \\ h=6.9\: ft \end{gathered}[/tex]Therefore, the height of the Twin tent at the center is 6.9 ft
Part 2:
The Mini tent will have edges 5 ft long.
The height of the tent is given by
[tex]h=a\cdot\frac{\sqrt[]{3}}{2}[/tex]Where a is the length of the edge of the triangle.
Since we are given that a = 5 ft
[tex]\begin{gathered} h=a\cdot\frac{\sqrt[]{3}}{2} \\ h=5\cdot\frac{\sqrt[]{3}}{2} \\ h=4.3\: ft \end{gathered}[/tex]Therefore, the height of the Mini tent at the center is 4.3 ft
Part 3:
The Family-Size tent will have a height of 10 ft at the center.
Recall that the height of the tent is given by
[tex]h=a\cdot\frac{\sqrt[]{3}}{2}[/tex]Re-writing the formula for edge (a)
[tex]a=h\cdot\frac{2}{\sqrt[]{3}}[/tex]Since we are given that h = 10 ft
[tex]\begin{gathered} a=h\cdot\frac{2}{\sqrt[]{3}} \\ a=10\cdot\frac{2}{\sqrt[]{3}} \\ a=\frac{20}{\sqrt[]{3}} \\ a=11.6\: ft \end{gathered}[/tex]Therefore, the length of edges of the Family-Size tent is 11.6 ft
For the polynomial below, 3 is a zero.f(x) = x^3+ 3x^2-11x-21Express f(x) as a product of linear factors.f (x) = ?
EXPLANATION
Given the polynomial f(x) = x^3 +3x^2 -11x -21
Separating the expression into groups as shown as follows:
If Triangle ABC is dilated by a scale factor of 3 and the length of side AB is 15 inches, what is the length of side A'B'? Complete the statement: The length of side A'B' would be inches. Your answer
If Triangle ABC is dilated by a scale factor of 3 and the length of side AB is 15 inches, what is the length of side A'B'? Complete the statement: The length of side A'B' would be inches.
To find out the length side of A'B' multiply the length side AB by the scale factor
so
A'B'=3*(15)=45 inches
This is not from a test or graded assessment. The Question is included in the picture.
Given:
[tex]\begin{gathered} g(x)=-x^5-4x^3+6x \\ \\ h(x)=x^4+2x^3-2x^2+x-7 \\ \\ j(x)=3x^4+7x^2 \end{gathered}[/tex]It's required to determine if the functions are odd, even, or neither.
An even function satisfies the property:
f(-x) = f(x).
And an odd function satisfies the property:
f(-x) = -f(x)
We substitute x by -x on each function as follows:
[tex]\begin{gathered} g(-x)=-(-x)^5-4(-x)^3+6(-x) \\ \\ g(-x)=x^5+4x-6x \end{gathered}[/tex]Note the function g(-x) is the inverse (negative) of g(x), thus,
g(x) is odd
Now test h(x):
[tex]\begin{gathered} h(-x)=(-x)^4+2(-x)^3-2(-x)^2+(-x)-7 \\ \\ h(-x)=x^4-2x^3-2x^2-x-7 \end{gathered}[/tex]Comparing h(-x) and h(x) we can see none of the properties are satisfied, thus:
h(x) is neither odd nor even
Let's now test j(x):
[tex]\begin{gathered} j(-x)=3(-x)^4+7(-x)^2 \\ \\ j(-x)=3x^4+7x^2 \end{gathered}[/tex]Since j(-x) and j(x) are equal,
j(x) is even
Classify each Polynomial by degree and number of terms.1. X^3 + 5x 2. X^2 - 2x - 1 3. 5x^4 4. 6x^5 - 3x^2 + 7x + 9 5. -11x - 5 6. 4x^2 + 10 7. 128. 9x^3 - x^2 + 6x - 1]9. -3x^5 + 6x^4 v- 8THESE ARE THE OPTIONS Degree Name using degree 0 Constant 1 Linear 2 quadratic 3 Cubic 4 quartic 5 quintic 6 6th degreeTHESE ARE ALSO THE OTHER OPTIONSTerms NAME USING # OF TERMS1, monomial 2 , binomial3 trinomial4 or more polynomial
WILL GIVE BRAINLYEST 200 PO9NTS SENCE IM GIVING EXTRA
The range of the data set increased by 12 and the median of the data set is at 50.
Range of a Data SetThe Range of a data set is the difference between the lowest and highest values.
The range is the difference between the smallest and highest numbers in a list or set. To find the range, first put all the numbers in order. Then subtract (take away) the lowest number from the highest. The answer gives you the range of the list.
The given data set is;
15, 17, 19, 19, 22, 23, 25, 27, 32, 34
The range of this data set will be
range = 34 - 15 = 19.
Assuming we add another age of 46, the range will become
range = 46 - 15 = 31
This will impact the range of the data by 31 - 19 = 12.
The range will be impacted by an increase with 12.
Median of a Data SetThe median is the value that's exactly in the middle of a dataset when it is ordered. It's a measure of central tendency that separates the lowest 50% from the highest 50% of values. The steps for finding the median differ depending on whether you have an odd or an even number of data points.
The data set given is
48, 63, 75, 40, 32, 52, 35, 68, 83, 40
We have to rearrange the data points first before finding the median;
32, 35, 40, 40, 48, 52, 63, 68, 75, 83.
The median of the data set will be the average between 48 and 52 which is 50.
the median of the data is 50.
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At the farmer’s market, Joan bought apples at $1.20 per pound, cherries for $2.00 per pound and pears for $0.80 per pound. She bought a total of 9 pounds of fruit for $11.00. Joan bought twice as many pounds of apples than cherries. Let A be the weight of the apples, C be the weight of the cherries, and P be the weight of the pears. Formulate a system of equations to determine how many pounds of each type of fruit were bought. Do Not Solve.
We have here a case in which we need to translate a problem into algebraic expressions to solve a problem, and we have the following information from the question:
• We have that Joan bought:
0. Apples at $1.20 per pound
,1. Cherries at $2.00 per pound
,2. Pears at $0.80 per pound
• We know that she bought a total of 9 pounds of fruit.
,• We also know that she spent $11.00 for the 9 pounds of fruit.
,• Joan bought twice as many pounds of apples than cherries.
We need to label weights as follows:
• Weight of apples ---> A
,• Weight of cherries ---> C
,• Weight of pears ---> P
Now to find a system of equations to determine the number of pounds of each type of fruit was bought, we can proceed as follows:
1. We know that if we multiply the price of the fruit per pound by the weight in pounds, we will obtain the amount of money Joan spent in total. Then we have:
[tex]1.20a+2.00c+0.80p=11.00\rightarrow\text{ \lparen First equation\rparen}[/tex]2. We also know that the total weight of the fruits was equal to 9 pounds. Then we can translate it into an algebraic expression as follows:
[tex]a+c+p=9\rightarrow(\text{ Second equation\rparen}[/tex]3. And we know that Joan bought twice as many pounds of apples than cherries, and we can translate it as follows too:
[tex]\begin{gathered} 2a=c \\ \\ \text{ If we subtract c from both sides of the equation, we have:} \\ \\ 2a-c=c-c \\ \\ 2a-c=0\text{ \lparen Third equation\rparen} \end{gathered}[/tex]Now we have the following equations:
[tex]\begin{gathered} 1.20a+2.00c+0.80p=11.00 \\ \\ \begin{equation*} a+c+p=9 \end{equation*} \\ \\ \begin{equation*} 2a-c=0 \end{equation*} \end{gathered}[/tex]Therefore, we have that the correct option is the first option:
• 1.20a + 2.00c + 0.80p = 11.00
• a + c + p = 9
,• 2a - c = 0
[First option].