A. There are 17 students that are majoring in both english and mathematics
B. There are 139 students that are majoring in mathematics only
Explanation:Given that 274 are majoring in english, n(E) = 274
156 are majoring in mathematics, n(M) = 156
422 are majoring in at least one of english and mathematics, n(E U M) = 422
We have:
A. The number of students that are majoring in both english and mathematics is obtained as follows:
[tex]\begin{gathered} n(E\cap M)=n(E)+n(M)-n(E\cup M) \\ \\ =274+165-422 \\ \\ =17 \end{gathered}[/tex]B. The number of students that are majoring in mathematics only is obtained as follows:
[tex]\begin{gathered} n(M\text{ only\rparen=}n(M)-n(M\cap E) \\ \\ =156-17 \\ =139 \end{gathered}[/tex]Question 2: Construct a perpendicular to
AB at A and at B (Hint: Extend AB)
Answer:
Explanation:
Here, we want to construct a perpendicular line at the points
The steps are as follows:
a) We extend the line through A and B
b) Place the compass at Point A, and draw a small semi-circle. Now, divide this semi-circle into 2. The line drawn will be perpendicular to AB and it will pass through A
c) Repeat the same process for B
We have the sketch as follows:
Use the data set to determine which statements are correct. Check all that apply. 35, 41, 18, 75, 36, 21, 62, 29, 154, 70 The median is 36.The median is 38.5.There is an outlier.The lower quartile is 29. The lower quartile is 18. The upper quartile is 29.The upper quartile is 70. The interquartile range is 41.
Q1 = 35.75
Q2 = 40
Q3= 45.5
IQR = 9.75
Lower Outlier =15
Upper Outlier=55
1) Let's calculate the quartiles, by using a formula for that and considering that the Distributions is:
2) But we need to orderly write this distribution, so:
15 29 29 35 35 36 36 37 38 40 40 42 44 45 45 47 51 52 52 55
The first Quartile is given by the formula, below where n is the number of observations in this case, since we have a decimal let's find the average between the 5th and the 6th number:
[tex]\begin{gathered} Q_1=\frac{1}{4}(n+1)^{th} \\ Q_1=\frac{1}{4}(20+1)^{th} \\ Q_1=\frac{1}{4}(21)^{th} \\ Q_1=5.25 \\ Q_{,1}=\frac{35+36}{2}=35.75 \end{gathered}[/tex]Then The upper Quartile:
[tex]\begin{gathered} Q_3=\frac{3}{4}(n+1)^{th} \\ Q_3=\frac{3}{4}(21)^{th} \\ Q_3=\text{ 15.75 position} \\ Q_3=\frac{45+47}{2}=45.5 \end{gathered}[/tex]3) And the Second Quartile is going to be the median
[tex]Q_2=\text{ Median =}\frac{40+40}{2}=40[/tex]The interquartile range is going to be the difference, between the first quartile and the third one
IQR = 45.5 -35.75 =9. 75
The outliers in the distribution
15 29 29 35 35 36 36 37 38 40 40 42 44 45 45 47 51 52 52 55
They can be found by a formula:
[tex]\begin{gathered} Lower\colon Q_1-(1.5\text{ }\times IQR) \\ \text{Lower: 35.75-(1.5}\times9.75) \\ L=35.75-(14.625) \\ L=21.125\approx21 \\ \\ \text{Upper: Q}_3+(1.5\times IQR) \\ \text{Upper: }45.5+(1.5\times9.75) \\ \text{Upper: }60.125\approx60 \end{gathered}[/tex]The lower outlier is below 21.125, and the upper one 60.125 so in our distribution, Lowe Outlier is 15, and the Upper one, is closer to 60.125 in this case, 55.
The answers are:
Q1 = 35.75
Q2 = 40
Q3= 45.5
IQR = 9.75
what is 10 / 2 / 3 / 4 * 10 * 0 - 5 -10 + 20 * 10 + 10 divided by 10 Champs * 10
Given:
10÷2÷3÷4 x 10 x 0 - 5 - 10 + 20 x 10 + 10 ÷ 10 x 10
To simplify the problem above, use PEDMAS theorem.
Thus, we have:
Which sentence of transformation could be used to show the congruence between the triangles? I’m not sure if the answer is A,B,C or D? Some help would be nice
Answer:
a translation of 1 unit to the right and 3 units up and then a reflection across the y-axis
Explanation:
First, let's identify the coordinates of the vertex R, S, T and its images R', S', and T'.
R(-6, -2) ---> R'(5, 1)
S(-5, -5) ---> S'(4, -2)
T(-3, -3) ---> T'(2, 0)
Then, we can observe that the transformation was a translation of 1 unit to the right and 3 units up and the a reflection over the y-axis because:
A translation of 1 unit right and 3 units up is made by the following rule
(x, y) ---> (x + 1, y + 3)
So, each vertex is translated to
R(-6, -2) ---> (-6 + 1, -2 + 3) = (-5, 1)
S(-5, -5) ---> (-5 + 1, -5 + 3) = (-4, -2)
T(-3, -3) ---> (-3+ 1, -3 + 3) = (-2, 0)
Then, the reflection over the y-axis is
(x, y) ---> (-x, y)
So,
(-5, 1) ---> R'(5, 1)
(-4, -2) ---> S'(4, -2)
(-2, 0) ---> T'(2, 0)
Therefore, the answer is:
a translation of 1 unit to the right and 3 units up and then a reflection across the y-axis
[tex]2x {}^{2} + 2x - 4 = 0[/tex]Find zeros/roots by completing the sqaures
Answer:
x = 1 and -2
Explanation
Given the expression
2x^2 + 2x - 4 = 0
We are to find the zero of the equation using the completing the square method
Step 1: Divide through by 2
2x^2/2 + 2x/2 - 4/2 = 0/2
x^2 + x - 2 = 0
Step 2: Add 2 to both sides of the equation
x^2 + x - 2 + 2 = 0+2
x^2 + x = 2
Step 3: Complete the square by adding the square of half of coefficient of x to both sides as shown
Coefficient of x is 1
Half of 1 = 1/2
Square of 1/2 = (1/2)^2 = 1/4
Add 1/4 to both sides
x^2 + x + (1/2)^2= 2 + 1/4
(x+1/2)^2 = 9/4
Sqare root both sides
x + 1/2 = \sqrt[9/4]
x + 1/2 = 3/2
x = 3/2 - 1/2 and -3/2 - 1/2
x = 2/2 and -4/2
x = 1 and -2
Anyone willing to help me? i’ll give 17 points
Find the x-and y-intercepts of the graph of x - 2y = 32. State each answer as an integer or an improper fraction in simplest form.
The x- intercepts of the function is (32,0) and y- intercepts of the function is (0, -16).
Given,
In the question:
The equation is :
x - 2y = 32
To find the x-and y-intercepts of the graph.
Now, According to the question;
Equation is :
x - 2y = 32
Find x - intercepts
Isolate the dependent variable:
y = x/2 - 16
Find the x - intercept of the function
x = 32
Equation is :
x - 2y = 32
Find y - intercepts
Isolate the dependent variable:
y = x/2 - 16
Find the y - intercept of the function
y = -16
Hence, The x- intercepts of the function is (32,0) and y- intercepts of the function is (0, -16).
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[tex]4 \sqrt{5} (3 \sqrt{5} + 8 \sqrt{2} )[/tex]how do u do this
we have
[tex]4\sqrt{5}(3\sqrt{5}+8\sqrt{2})[/tex]Apply distributive property
[tex]4\sqrt{5}(3\sqrt{5})+4\sqrt[]{5}(8\sqrt{2})[/tex][tex]12\sqrt[]{25}+32\sqrt[]{10}[/tex]simplify
[tex]\begin{gathered} 12(5)+32\sqrt[]{10} \\ 60+32\sqrt[]{10} \end{gathered}[/tex]This is solving rational equationsI really need some help. Please explain how you get each step if u can.
We are given the following equation:
[tex]\frac{-4}{x+4}=-\frac{3}{x+6}[/tex]we are asked to determine any extraneous solutions. To do that we will determine the values of "x" that solve the equation. First, we will cross multiply the equation:
[tex]-4(x+6)=-3(x+4)[/tex]Now, we can multiply by -1, we get:
[tex]4(x+6)=3(x+4)[/tex]Now we use the distributive property on both sides, we get:
[tex]4x+24=3x+12[/tex]Now, we subtract "3x" from both sides:
[tex]\begin{gathered} 4x-3x+24=3x-3x+12 \\ x+24=12 \end{gathered}[/tex]Now we subtract 24 from both sides:
[tex]\begin{gathered} x+24-24=12-24 \\ x=-12 \end{gathered}[/tex]Therefore, the solution is x = -12. The extraneous solutions are the solutions that are not in the domain of the original function.
In the domain of the original function, we have that the values of:
[tex]\begin{gathered} x=-4 \\ x=-6 \end{gathered}[/tex]Would make the denominators equal to zero, and therefore, they are not in the domain. Since the solution is none of these values there are no extraneous solutions.
The point enter your response here is also on the graph of the equation.
An equation that has a graph that is symmetric to the origin, has a reflection of each point through the origin, reflects across the x-axis and y-axis.
Therefore, if one point of the graph is (-4,1).
Reflected to both axis, we can find that another is is (-x,-y) = (-(-4), -1) = (4, -1)
So, the answer is the point (4, -1)
Suppose you go to a conference attended by 20 Canadians and 20 Americans. How many people must you meet to be certain that you have met two Americans?
There are 20 Canadians and 20 Americans at the conference.
we met the first person may be Canadian or American
Case 1:
If the first person is Canadian
We meet the second person
The second person also maybe Canadian or American
If all the first 20 peoples are Canadian
The next two people should be American
Hence we should meet a minimum of 22 people.
Case 2:
If the first person is American
We meet the second person
The second person also maybe Canadian or American
If all the next 20 peoples are Canadian
Then the 22nd people should be American
Hence we should meet a minimum of 22 people.
Case 3:
If the first person is American
We meet the second person
The second person also maybe Canadian or American
If the second people is also American
Hence we met 2 Americans in two attempts.
Result:
We need to meet 22 people to meet two americans.
Given the explicit formula below, which is the appropriate list of the first 4 numbers:An = 3(2)n-1A2, 6, 18, 54, .B3, 6, 9, 12,..С3, 6, 12, 24, ...D1, 3, 5, 7, ...
The simpliest way to answer the question is by iteration, substituting n with the numbers 1, 2, 3, 4 to find the first 4 numbers. So we have the formula;
[tex]A_n=3(2)^{n-1}[/tex]When n = 1,
[tex]\begin{gathered} A_n=3(2)^{n-1} \\ A_1=3(2)^{1-1} \\ A_1=3(2)^0 \\ A_1=3 \end{gathered}[/tex]When n = 2,
[tex]\begin{gathered} A_n=3(2)^{n-1} \\ A_2=3(2)^{2-1} \\ A_2=3(2)^1 \\ A_2=6^{} \end{gathered}[/tex]When n = 3,
[tex]\begin{gathered} A_n=3(2)^{n-1} \\ A_3=3(2)^{3-1} \\ A_3=3(2)^2 \\ A_3=12 \end{gathered}[/tex]When n = 4,
[tex]\begin{gathered} A_n=3(2)^{n-1} \\ A_4=3(2)^{4-1} \\ A_4=3(2)^3 \\ A_4=24 \end{gathered}[/tex]Therefore when n = {1, 2, 3, 4}, An = {3, 6, 12, 24}, making 3, 6, 12, 24 the first 4 numbers of our formula.
Therefore the answer is LETTER C.
Points L,M and N are collinear.M is between L and N. You are given LM=13 and LN=20.Find the length of MN
ANSWER
MN = 7
EXPLANATION
Let's draw a diagram first:
Since all the points are collinear we can use the segment addition postulate:
[tex]LM+MN=LN[/tex]Replacing with the values we know:
[tex]13+MN=20[/tex]And solving for MN:
[tex]MN=20-13=7[/tex]We have that the length of segment MN is 7.
Knowledge check that with the kids on the other hand ✋ the
. The function h measures the height.
. The units on h meters.
. h'(t) measures the velocity
. The units on h'(t) are meters per second
. The units on t are seconds
What are the coordinate points of I if it is halfway between points F (2, 3)and X (10,9) on FX?
ANSWER
EXPLANATION
If point I is halfway between points F and X on the line FX, then point I is the midpoint of this line. Its coordinates are given by half the distance in each direction - the x and y-directions, betwen points F and X,
identify the placevalue for each digit in the number 95.26
The place value of;
9 is 9 tens
5 is 5 unit
2 is 2 tenth
6 is 6 hundredth
Perform the following mathematical operation and report the answer to the appropriate number of significant figures.
We know the least precise place value is in the 10's place.
67.4 +43 +30 + 42.10 = [?]
The least precise place value is in the 10's place is 182.5.
Given that, 67.4 +43 +30 + 42.10.
What are decimal numbers?Decimals are one of the types of numbers, which has a whole number and the fractional part separated by a decimal point. The dot present between the whole number and fractions part is called the decimal point.
Significant figures are the number of digits in a value, often a measurement, that contribute to the degree of accuracy of the value.
Now,
67.4 + 42.10 + 73
= 109.5 + 73
= 182.5
Hence, the least precise place value is in the 10's place is 182.5.
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he spent $54integer:
Since the person is spending it means that the money they have is decreasing.
when anything is decreasing we use the negative sign
[tex]He\text{ spent \$54}\rightarrow-54[/tex]If the cost of a loaf of bread is now $2.75 and is increasing at 5% per year, what will it cost 10 years from now. Write an exponential equation for this scenario then use it to solve the problem
To estimate the cost (C) after t years with an increasing rate r, use the formula below:
[tex]C(t)=C_0*\left(1+r\right)^t[/tex]
In this question:
C0 = initial cost = $2.75
r = rate = 0.05
t = time = 10 years
Substituting the values in the equation:
[tex]\begin{gathered} C(10)=2.75*\left(1+0.05\right)^{10} \\ C(10)=2.75*1.05^{10} \\ C(10)=2.75*1.629 \\ C(10)=4.48 \end{gathered}[/tex]Answer:
Equation:
[tex]\begin{gathered} C(t)=2.75(1+0.05)^t \\ C(10)=2.75(1+0.05)^{10} \end{gathered}[/tex]Cost: C(10) = $4.48.
Given the following piecewise function, evaluate f(-1).f(x) =3x +4. X < -1-8x +8 x> -1
We have the expression:
[tex]3x+4\text{ if x}\leq-1[/tex]So, when x=-1 we need to use this expression, so lets plot -1 on it:
[tex]3(-1)+4=-3+4=1[/tex]so f(-1)=1.
Select the correct answer.A local magazine, available by subscription and at newsstands, mailed a comment card to each of its subscribers. It asked the readers how satisfied they were with the magazine's coverage of local events and interests. Based on the information received from the comment cards, the magazine created a television ad saying that 97% of its readers were extremely satisfied with the content of the magazine.Why might this sample be biased?A. The magazine changed its content for the issue including the comment card.B. Very few comment cards were returned.C. The survey only considered subscribers, and not those who purchase the magazine at newsstands.D. Not enough comment cards were distributed.
Hello there. To solve this question, we'll analyze each of the options and determine why it is the case for this sample to be biased.
We know that this local maganize is available by subscription and at newsstands and they asked the readers by mailing them comment cards for the subscribers.
A. The magazine changed its content for the issue including the comment card.
Since the question didn't mention this fact, we cannot assume this to be a correct option, we only know they sent comment card to the readers.
B. Very few comment cards were returned
The question says that 97% of the readers were extremely satisfied with the content of the magazine. But we know that these readers that received a comment card were subscribers.
This means that, considering each subscriber sent its comment card back, 3% of them were not satisfied, but we don't know how many subscribers this magazine has.
Hence this answer is not correct as well.
C. The survey only considered subscribers, and not those who purchase the magazine at newsstands.
This might be the answer, considering that the comment card were only mailed to the subscribers and, since we don't know the proportion between subscribers and those who buy at the newsstands, this sample is biased.
D. Not enough comment cards were distributed.
The question said that the magazine mailed a comment card for each of its subscribers, so we cannot take this option into consideration.
Which of the following is true with respect to the following functions:f(x) = 3] x + 14 | g(x)= x4 + 3x2 - 14h(x) = 3% +1i(x) = log2 (x + 1)
To answer the question, let us plot the graphs of all the functions.
f(x):
[tex]f(x)=3|x+14|[/tex]g(x):
[tex]g(x)=x^4+3x^2-14[/tex]h(x):
[tex]h(x)=3^x+1[/tex]i(x):
[tex]i(x)=\log _3(x+1)[/tex]From the graphs shown above, it can be seen that the range of the function i(x) goes from -∞ to +∞. This is the function with the greatest negative range.
Therefore, the correct option is OPTION D
What is money? 1. A store of value2. A medium of exchange3. A measure of valuea. Money simplifies the exchange process because it’s a means of indicating how much something costs.b. To use money to buy the goods and services you want.c. People are willing to hold onto it because they’re confident that it will keep its value over time.it is math even if it doesnt look like it
Given data:
Money can be defined a medium of exchange.
Thus, money is juat a medium of echnage.
can you help me on this one?I need to determine whether the figure is a parallelogram using the distance formula.
If we graph the given points, we have:
One property of the parallelograms is that their opposite sides are equal.
Then, we have to verify if the segments QT and RS are equal.
[tex]QT=RS[/tex]To find the measure of segments QT and RS, we can use the distance formula.
[tex]\begin{gathered} d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}\Rightarrow\text{ Distance formula} \\ \text{ Where} \\ (x_1,y_1)\text{ and }(x_2,y_2)\text{ are the coordinates of the points} \end{gathered}[/tex]• Measure of segment QT
[tex]\begin{gathered} (x_1,y_1)=Q(-10,-2) \\ (x_2,y_2)=T(-11,-8) \\ d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=\sqrt[]{(-11-(-10))^2+(-8-(-2))^2} \\ d=\sqrt[]{(-11+10)^2+(-8+2)^2} \\ d=\sqrt[]{(-1)^2+(-6)^2}\rbrack \\ d=\sqrt[]{1+36} \\ d=\sqrt[]{37} \\ d\approx6.08\Rightarrow\text{ The symbol }\approx\text{ is read 'approximately'} \end{gathered}[/tex]• Measure of segment RS
[tex]\begin{gathered} (x_1,y_1)=R(1,-1) \\ (x_2,y_2)=S(1,-7) \\ d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=\sqrt[]{(1-1)^2+(-7-(-1))^2} \\ d=\sqrt[]{(0)^2+(-7+1)^2} \\ d=\sqrt[]{0+(-6)^2} \\ d=\sqrt[]{(-6)^2} \\ d=\sqrt[]{36} \\ d=6 \end{gathered}[/tex]As we can see, the segments QT and RS are different.
[tex]\begin{gathered} QT\ne RS \\ 6.08\ne6 \end{gathered}[/tex]Then, the figure does not satisfy the mentioned property of parallelograms.
Therefore, the figure is not a parallelogram.
y varies directly as z and inversely as x; write the sentence as an equation
Given:
(y) varies directly as (z) and inversely as (x):
[tex]\begin{gathered} y\propto z;y\propto\frac{1}{x} \\ \\ so,y\propto\frac{z}{x} \end{gathered}[/tex]so, the equation will be:
[tex]y=\frac{kz}{x}[/tex]Where: k is the proportionality constant
You need to borrow money for gas, so you ask your mother and your sister. You can only borrow money from one of them. Before giving you money, they each say theywill make you play a game. Your sister says she wants you to spin a spinner with six outcomes, numbered 1 through 6, on it. She will give you $3 times the number thatthe spinner lands on. Your mother says she wants you to spin a spinner with two outcomes, blue and red, on it. She will give you $9 if the spinner lands on blue and $21 ifthe spinner lands on red. Determine the expected value of each game and decide which offer you should take.AnswerKeypadKeyboard ShortcutsExpected value for your sister's game:Expected value for your mother's game: SWhich offer should you take?your sister's offeryour mother's offer
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
expected value of each game = ?
Step 02:
Expected value:
Sister:
total outcomes = 6
1 - 6
probability = 1/6
1 | 2 | 3 | 4 | 5 | 6
$3 | $3 | $3 | $3 | $3 | $3
1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6
[tex]\text{expected value = \$3}\cdot\frac{1}{6}+\text{ \$3}\cdot\frac{1}{6}+\text{ \$3 }\cdot\text{ }\frac{1}{6}\text{ + \$3}\cdot\frac{1}{6}+\text{ \$3}\cdot\frac{1}{6}\text{ + \$3 }\cdot\text{ }\frac{1}{6}[/tex]expected value (sister) = $3
Mother:
total outcomes = 2
blue - red
probability = 1/2
blue | red
$9 | $21
1/2 | 1/2
[tex]\text{expected value = \$9 }\cdot\text{ }\frac{1}{2}\text{ + \$21}\cdot\frac{1}{2}[/tex]expected value (mother) = $15
The answer is:
Expected value (sister) = $3
Expected value (mother) = $15
Mother's offer.
Lucy earns $326.87 each week. The federal government withholds 18% ofthat for federal income tax. How much is withheld from her pay annuallyfor federal income tax?a. $58.84b. $2,967.05c. $13,937.74d. $3,059.50
ANSWER:
d. $3,059.50
STEP-BY-STEP EXPLANATION:
The first thing is to calculate the annual salary, knowing that a year has a total of 52 weeks, therefore:
[tex]\begin{gathered} a=326.87\cdot52 \\ a=16997.24 \end{gathered}[/tex]The annual price is $ 16,997.24, we calculate 18% of this value, as follows:
[tex]\begin{gathered} tax=16,997.24\cdot\frac{18}{100} \\ tax=3059.50 \end{gathered}[/tex]Which means that federal income tax is $ 3,059.50
Consider the first few terms as well as the last few terms of the sum. Find a way to simplify and use your observation to evaluate the sum. Write the exact answer. Do not round.
The first three terms of the sum are:
[tex]\begin{gathered} \frac{1}{3}-\frac{1}{3+1}=\frac{1}{3}-\frac{1}{4}\to\text{first} \\ \frac{1}{4}-\frac{1}{4+1}=\frac{1}{4}-\frac{1}{5}\to\text{second} \\ \frac{1}{5}-\frac{1}{6}\to\text{ third} \end{gathered}[/tex]On the other hand, the three last terms of the sum are
[tex]\begin{gathered} \frac{1}{10}-\frac{1}{10+1}=\frac{1}{10}-\frac{1}{11}\to\text{tenth} \\ \frac{1}{11}-\frac{1}{12}\to\text{ eleventh} \\ \frac{1}{12}-\frac{1}{13}\to\text{twelfth} \end{gathered}[/tex]Notice that there is a -1/4 in the first term, a 1/4 in the second term, a 1/5 in the second term, and 1/5 in the third term, and so on. Once we add those factors, the result will be zero. Thus, the result of the sum is equal to the first number in the first term minus the second number in the twelfth term.
[tex]\begin{gathered} \sum ^{12}_{i=3}(\frac{1}{i}-\frac{1}{i+1})=\frac{1}{3}-\frac{1}{13}=\frac{13-3}{39}=\frac{10}{39} \\ \Rightarrow\sum ^{12}_{i=3}(\frac{1}{i}-\frac{1}{i+1})=\frac{10}{39} \end{gathered}[/tex]The answer is 10/39
Write an equation for the line that passes through the given point and is perpendicular to the graph of the given equation y=-2x-1; (2, -1)
The product of the slopes of the perpendicular lines = -1
If the slope of one is m, then the slope of the other is -1/m
The given equation is y = -2x - 1
The form of the equation is y = m x + b, where
m is the slope of the line
So the slope of the given line is m = -2
To find the slope of the perpendicular line reciprocal it and change its sign
The slope of the perpendicular line = 1/2
Substitute it in the form of the equation
y = 1/2 x + b
To find b substitute x and y of the equation by the coordinates of any point on the line
The line passes through point (2, -1), then
x = 2 and y = -1
-1 = 1/2 (2) + b
-1 = 1 + b
Subtract 1 from both sides to find b
-1 - 1 = 1 - 1 + b
-2 = b
The equation of the perpendicular line is
y = 1/2 x - 2
[tex]y=\frac{1}{2}x-2[/tex]This picture is the paragraph of information to answer the questions. The second picture is the questions
Question 1
a) It could not represent the scenario because the vertex of the parabola is located at (-5,9), and x=-5 is a region beyond the bank.
b) This function does not fit the scenario because the parabola opens up, as if the fish fell towards the sky.
c) This function does not fit the scenario because the expression is negative for all values of x, which means that the fish always remains under water.
Question 2
The x-value at the center of the boat is 5.
Question 3
The fish jumps 4 feet high.
Question 4
The zeros of the function are x=3 and x=7 and they represent the locations over the x-axis where the fish comes out of the water and re-enters the river.
Question 5
The fish comes out of the water 2 feet away from the center of the boat.
Question 6
The domain is: 0≤x≤50.
The range is: -2021≤y≤4.
Question 7
a) The path of the fish is increasing at the interval [0,5).
b) The path of the fish is decreasing at the interval (5,50]
Question 8
a) The fish swims under water at the intervals [0,3) and (7,50].
b) The fish swims abov the water at the interval (3,7).
Explanation:Question 1:
Write the functions in vertex form.
a)
[tex]\begin{gathered} F\left(x\right)=-x^2-10x-16 \\ =-\left(x+5\right)^2+9 \end{gathered}[/tex]The vertex is located at (-5,9), but the bank is the line x=0 and the region x<0 corresponds to land. The fish cannot swim on the land, so it would be impossible for the fish to reach that point. Additionally, the function is negative for all positive values of x, so in the region that corresponds to the river, the fish never comes out of the water.
b)
[tex]F\left(x\right)=x^2-6x+13[/tex]Since the coefficient of the quadratic term is positive, the parabola opens up. Then, this trajectory corresponds to an object that "falls upwards", which is not possible.
c)
[tex]\begin{gathered} F\left(x\right)=-x^2+6x-13 \\ =-\left(x-3\right)^2-4 \end{gathered}[/tex]Notice that the function is negative for all values of x, which can be interpreted as if the fish never came out of the water, which does not correspond to the described situation.
Question 2:
Write the expression that describes the trajectory of the fish in vertex form. The x-coordinate of the vertex corresponds to the center of the boat.
[tex]\begin{gathered} F\left(x\right)=-x^2+10x-21 \\ =-\left(x-5\right)^2+4 \end{gathered}[/tex]The vertex of the parabola is (5,4), so the center of the boat is located at x=5.
Question 3:
The maximum height of the fish corresponds to the y-coordinate of the vertex. So, the fish's maximum height is y=4 (4 feet).
Question 4:
Set F(x)=0 and solve for x:
[tex]\begin{gathered} F\left(x\right)=0 \\ \Rightarrow-\left(x-5\right)^2+4=0 \\ \Rightarrow4=\left(x-5\right)^2 \\ \Rightarrow\left(x-5\right)^2=4 \\ \Rightarrow x-5=±\sqrt{4} \\ \Rightarrow x-5=±2 \\ \Rightarrow x=5±2 \\ \therefore x_1=3,x_2=7 \\ \end{gathered}[/tex]Then, the zeros of the function are x=3 and x=7, they represent the horizontal location at which the fish is located at the surface of the river, so they are the points where the fish comes out of the water and re-enters the water.
Question 5:
The points x=3 and x=7 are both 2 units away from x=5. Then, the fih comes out of the water and re-enters the water 2 feet away from the center of the boat.
Question 6:
a)
The domain corresponds to all the values of x that the function can take. Since the river is 50 feet wide and the bank is the y-axis, then, the river covers the region 0≤x≤50, which is the same as the domain.
b)
The range corresponds to all the values over the y-axis that the function can take when evaluated at values from the domain.
To find the range, we have to find the maximum and minimum values of F(x) for 0≤x≤50. Since the maximum value of the function is 4 (the maximum height), just check for the values of F(0) and F(50) to find the possibilities for the minimum:
[tex]\begin{gathered} F\left(0\right)=-\left(0-5\right)^2+4 \\ =-25+4 \\ =-21 \end{gathered}[/tex][tex]\begin{gathered} F\left(50\right)=-\left(50-5\right)^2+4 \\ =-\left(45\right)^2+4 \\ =-2025+4 \\ =-2021 \end{gathered}[/tex]Then, the minimum value of the function F for the values in the domain is -2021. Therefore, the range is: -2021≤y≤4.
Question 7:
The path of the fish increases until it reaches it maximum point at x=5, then it decreases. Then:
a) The path of the fish increases at 0≤x<5.
b) The path of the fish decreases as 5.
Question 8:
The fish is initially under water, it comes out at the first zero of the function x=3, it travels through the air until it re-enters the water at x=7 and it continues under water from that point on. Then:
a) The fish is under water in the interval 0≤x<3 and 7.
b) The fish is above the water in the interval 3.