Given the equation of the line below,
[tex]8y-16=5x[/tex]If the line passes through the point,
[tex](5,-5)[/tex]Re-writing the eqaution of the line in slope intercept form,
[tex]\begin{gathered} 8y-16=5x \\ 8y=5x+16 \\ \text{Divide both sides by 8} \\ y=\frac{5x}{8}+\frac{16}{2} \\ y=\frac{5}{8}x+2 \end{gathered}[/tex]The slope of the perpendicular line is the negative reciprocal of the slope of the eqaution of the line in the slope-intercept form given above
The general form of the slope-intercept form of the equation of a straight line is,
[tex]\begin{gathered} y=mx+c \\ \text{Where m is the slope} \\ y=\frac{5}{8}x+2 \\ m=\frac{5}{8} \\ \text{Slope of the perpendicular line is} \\ m_1=-\frac{1}{m} \\ m_{1_{}}=-\frac{1}{\frac{5}{8}}=-1\times\frac{8}{5}=-\frac{8}{5} \end{gathered}[/tex]The formula to find the equation of a line with point (5, -5) below is,
[tex]\begin{gathered} \frac{y-y_1}{x-x_1}=m_1 \\ \text{Where} \\ (x_1,y_1)=(5,-5) \\ m_1=-\frac{8}{5} \end{gathered}[/tex]Substitute the values into the formula of the eqaution of a straight line,
[tex]\begin{gathered} \frac{y-(-5)}{x-5}=-\frac{8}{5} \\ \frac{y+5}{x-5}=-\frac{8}{5} \\ \text{Crossmultiply} \\ 5(y+5)=-8(x-5) \\ 5y+25=-8x+40 \\ \text{Collect like terms} \\ 5y=-8x+40-25 \\ 5y=-8x+15 \\ \text{Divide both sides by 5} \\ \frac{5y}{5}=-\frac{8}{5}x+\frac{15}{5} \\ y=-\frac{8}{5}x+3 \end{gathered}[/tex]Hence, the right option is C
Describe a situation that could be represented by theequation y=x-0.3x.Be sure to explain what x and y mean in your situation,
We are asked to describe a situation that could be represented by the equation
[tex]y=x-0.3x[/tex]Suppose that y is the number of liters of water in a tank.
And x is the number of hours.
Each hour, 30% (0.3) of the water is evaporated from the tank. (subtracted)
So the equation completely models the above scenario.
[tex]y=x-0.3x[/tex]For example:
What will be the amount of water in the tank after 10 hours?
[tex]undefined[/tex]Two shaded cubes are shown. 6 feet 4 feet 6 feet 4 feet 4 feet 6 feet Ben states that the combined volume of these two shaded cubes is equal to the volume of this cube. 10 feet Vo feet 10 feet Find the combined volume of the two shaded cubes, and use it to explain whether Ben is right or not. Ben is correct. Ben is not correct.
The volume of a cube is given by the formula:
[tex]V=s^3[/tex]Where
s is the side length of a cube
Combined Volume of two cubes
Now, let's find the combined volume of the cubes with side lengths shown:
[tex]\begin{gathered} V=4^3+6^3 \\ V=64+216 \\ V=280 \end{gathered}[/tex]Volume of larger cube
Plugging into the formula, we get:
[tex]\begin{gathered} V=s^3 \\ V=10^3 \\ V=1000 \end{gathered}[/tex]Thus, we clearly see that:
[tex]280\neq1000[/tex]Thus,
Ben isn't correct
Mason is standing on the seashore. He believes that if he makes a wish
and throws a seashell back into the ocean, his wish will come true. Mason is
standing at the origin of a coordinate plane and the shoreline is represented by the
graph of the line
y = 1.5x + 13. Each unit represents 1 meter. How far does Mason need to be able
to throw the seashell to throw one into the ocean? Round your answer to the
nearest centimeter.
To throw a seashell into the ocean, Mason needs to throw it above the line y = 1.5x + 13, which represents the shoreline.
This means that the seashell's height, y, must be greater than 1.5 times its horizontal distance, x, plus 13. We can use the Pythagorean theorem to find the distance, d, that Mason needs to throw the seashell, given by d = sqrt(x^2 + y^2).
We want to find the minimum value of d that satisfies y > 1.5x + 13. This occurs when y is equal to 1.5x + 13, since any larger value of y would require a larger value of d.
So we can substitute y = 1.5x + 13 into the equation for d and get:
d = sqrt(x^2 + (1.5x + 13)^2)
To find the minimum value of d, we can use calculus and find the derivative of d with respect to x, and set it equal to zero. Alternatively, we can use a graphing calculator or an online tool to plot the function d and find its minimum point. Either way, we get that the minimum value of d occurs when x is approximately -5.2 meters and y is approximately 5.2 meters. The corresponding value of d is approximately 14.7 meters.
Therefore, Mason needs to be able to throw the seashell at least 14.7 meters, or 1470 centimeters, to throw it into the ocean.
For the figure below, give the following. (a) one pair of vertical angles (b) one pair of angles that form a linear pair (c) one pair of angles that are congruent 1/2 1 3 4 5 / 6 7 78
Let's remember the following definitions:
1. Vertical angles are those angles that are opposite to each other and share the same vertex. Their measures are equal (They are congruent).
2. A Linear pair of angles are those adjacent angles formed when two lines intersect each other. They are Supplementary, which means that they add up to 180 degrees.
For this case you can see two lines "l" and "m" that are cut by the line "n".
a) Based on the definitions shown above, you can identify this pair of Vertical angles:
[tex]\angle1\text{ and }\angle4[/tex]Because the are opposite and share the same vertex.
b) You can also identify this pair of angles that form a Linear pair:
[tex]\angle2\text{ and }\angle4[/tex]c) Since you know that Vertical angles are congruent, you can determine that this pair of angles are Vertical angles and congruent:
[tex]\angle6\text{ and }\angle7[/tex]Therefore, the answers are:
a)
[tex]\angle1\text{ and }\angle4[/tex]b)
[tex]\angle2\text{ and }\angle4[/tex]c)
[tex]\angle6\text{ and }\angle7[/tex]7. Angela bought some sugar and strawberries to make strawberry jam.Sugar costs $1.80 per pound, and strawberries cost $2.50 per pound.Angela spent a total of $19.40. Which point on the coordinate plane couldrepresent the pounds of sugar and strawberries that Angela used to makejam?
Equation of the Line
Let's use the following variables:
x = pounds of sugar
y = pounds of strawberries
Angela spent a total of $19.40 to make strawberry jam, thus:
1.80x + 2.50y = 19.40
The equation of the line represents the relationship between x and y. Any point that solves the problem must lie in the line.
The image shows the graph of a line, but we need to be sure it represents the equation above. A small checkup will be done as follows:
For x = 0, solve for y:
1.80*(0) + 2.50y = 19.40
y = 19.4 / 2.5 = 7.76
This point corresponds to the y-intercept (0, 7.76). It can be correctly found on the graph.
For y = 0, solve for x:
1.80x + 2.50*(0) = 19.40
x = 19.40 / 1.8 = 10.78
This point corresponds to the x-intercept at (10.78, 0). It can also be found on the graph.
Now we are sure the line is the representation of our equation, the only point that lies on that line is B(8, 2).
If we substitute x = 8, y = 2:
1.80*(8) + 2.50*(2) = 19.40
14.4 + 5 = 19.40
19.40 = 19.40
The equation is satisfied, thus, the answer is:
Point B
sin data cos data tan datacsc date sec data cot data
P (7/25, 24/25)
Sin data= 24/25
Cos data = 7/25
Tan data= (24/25)/(7/25)= 24/7
Csc data= 1/(24/25)= 25/24
Sec data= 1/(7/25)= 25/7
Cotan data= 1/(24/7)= 7/24
Class Work...Exit Ticket... 11.25.2020 Malik picked forty-five oranges in five minutes. At this rate, how many oranges will she pick per minute. Classwork/Participation. 5 points
Malike picked 45 oranges in 5 minutes
Work = Rate x time
If she can pick 45 oranges in 5 minutes
Mathematically
45 oranges ========= 5minutes
x oranges ========== 1 minute
Introduce cross multiplication
45 * 1 = 5 * x
45 = 5x
Divide both sides by 5
45/5 = 5x/5
x = 45 / 5
x = 9 oranges
Malik can pick 9 oranges in 1 minute
The answer is 9 oranges
Using the given graph of the function f, find the following.(d) whether it is even, odd, or neither
The function is even function if graph of function is symmetric about y-axis, he function is odd function if graph of function is symmetric about origin.
From the graph of function it can be observed that (-2,0) and ()
blank +0=9 is a associative b. commutativec identity property
It is an identity property
Any number plus zero will give that number
f(x)= - 9x+2 Find the domain of the function. Type answer in interval notation.
ANSWER:
Domain: (-∞, ∞)
STEP-BY-STEP EXPLANATION:
We have the following function:
[tex]f(x)=-\:9x+2\:[/tex]The domain of a function is the set of all possible input values of the function. In this case, it would be the interval of values that x can take.
In this function, x can take any value in real numbers.
Therefore, in that case, it will be:
[tex]D=(-\infty,\infty)[/tex]what is 8 1/2 / 11 as a mixed number or fraction
Answer: 17/22
Step-by-step explanation:
Answer:
Step-by-step explanation:
Answer:
3.68181818182=33409090909150000000000
Showing the work
Rewrite the decimal number as a fraction with 1 in the denominator
3.68181818182=3.681818181821
Multiply to remove 11 decimal places. Here, you multiply top and bottom by 1011 = 100000000000
3.681818181821×100000000000100000000000=368181818182100000000000
Find the Greatest Common Factor (GCF) of 368181818182 and 100000000000, if it exists, and reduce the fraction by dividing both numerator and denominator by GCF = 2,
368181818182÷2100000000000÷2=18409090909150000000000
Simplify the improper fraction,
=33409090909150000000000
In conclusion,
3.68181818182=33409090909150000000000
h (t) = 2t - 2 g(t) = 4t + 5 Find (h(g(t))
ANSWER:
[tex]h(g(t))=8t-8[/tex]STEP-BY-STEP EXPLANATION:
We have the following functions:
[tex]\begin{gathered} h(t)=2t-2 \\ g(t)=4t+5 \end{gathered}[/tex]To calculate h (g (t)) we must do the following:
[tex]\begin{gathered} h\mleft(g\mleft(t\mright)\mright)=2\cdot(4t+5)-2 \\ h(g(t))=8t+10-2 \\ h(g(t))=8t-8 \end{gathered}[/tex]Which expression has a quotient of 63? 1) 4650÷752) 2867÷473) 3276÷52
To find the term with quotient in 63 in from the given option divide each quotient with 63. If the quatiend is divisible that term contain the quotient 63.
The quotient in the first option is 4650. Divide the quotient with 63.
[tex]\frac{4650}{63}=72.38[/tex]The final answer contains decimal places. Thus, there first option does not contain 63 as quotient.
The quotient in the second term is 2867. Divide the quotient with 63.
[tex]\frac{2867}{63}=45.190[/tex]The final answer contains decimal places. Thus, there second option does not contain 63 as quotient.
The quotient in the second term is 3276. Divide the quotient with 63.
[tex]\frac{3276}{63}=52[/tex]The final answer does not contain any decimal places. Thus, the third option contains 63 as quotient.
Thus, the correct option is option 3) 3276÷52.
solve quadratic formulax^2-4x+3=0
The general formula for a equation of the form:
[tex]ax^2+bx+c=0[/tex]is:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]In this case we notice that a=1, b=-4 and c=3. Plugging this values in the general formula we get:
[tex]\begin{gathered} x=\frac{-(-4)\pm\sqrt[]{(-4)^2-4(1)(3)}}{2(1)} \\ =\frac{4\pm\sqrt[]{16-12}}{2} \\ =\frac{4\pm\sqrt[]{4}}{2} \\ =\frac{4\pm2}{2} \end{gathered}[/tex]then:
[tex]x_1=\frac{4+2}{2}=\frac{6}{2}=3[/tex]and
[tex]x_2=\frac{4-2}{2}=\frac{2}{2}=1[/tex]Therefore, x=3 or x=1.
Triangle is rotated 180° around the origin. What will be the coordinates for Triangle J'K'L'? A(6,7)(6,2)(3,7)B(7,-6)(2,-7)(-3,-7)C(-6,-7)(-6,-2)(-3,-7)D(-7,6)(2,-6)(-7,3)
Answer:
A. (6,7)(6,2)(3,7)
Explanation:
From the graph, the coordinates of J, K and L are:
[tex]J(-6,-7),K(-6,-2)\text{ and L}(-3,-7)[/tex]When a point (x,y) is rotated 180° around the origin, we have the transformation rule:
[tex](x,y)\to(-x,-y)[/tex]Therefore, the coordinates for Triangle J'K'L' are:
[tex]J^{\prime}(6,7),K^{\prime}(6,2)\text{ and L'}(3,7)[/tex]The correct choice is A.
Rain equation of a hyperbola given the foci and the asymptotes
The equation for a hyperbola that opens up and down has the following general form:
[tex]\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1[/tex]Where the foci of the hyperbola are located at (h,k+c) and (h,k-c) with c given by:
[tex]c^2=a^2+b^2[/tex]And asymptotes with slopes given by a/b and -a/b.
The hyperbola with the equation that we have to find has these two foci:
[tex](3,2-\sqrt{26})\text{ and }(3,2+\sqrt{26})[/tex]This means that:
[tex]\begin{gathered} (h,k-c)=(3,2-\sqrt{26}) \\ (h,k+c)=(3,2+\sqrt{26}) \end{gathered}[/tex]So we get h=3, k=2 and c=√26.
The slope of the asymptotes have to be 5 and -5 which means that:
[tex]\frac{a}{b}=5[/tex]Using the value of c we have:
[tex]c^2=26=a^2+b^2[/tex]So we have two equation for a and b. We can take the first one and multiply b to both sides:
[tex]\begin{gathered} \frac{a}{b}\cdot b=5b \\ a=5b \end{gathered}[/tex]And we use this in the second equation:
[tex]\begin{gathered} 26=(5b)^2+b^2=25b^2+b^2 \\ 26=26b^2 \end{gathered}[/tex]We divide both sides by 26:
[tex]\begin{gathered} \frac{26}{26}=\frac{26b^2}{26} \\ b^2=1 \end{gathered}[/tex]Which implies that b=1. Then a is equal to:
[tex]a=5b=5\cdot1=5[/tex]AnswerNow that we have found a, b, h and k we can write the equation of the hyperbola. Then the answer is:
[tex]\frac{(y-2)^2}{5^2}-\frac{(x-3)^2}{1^2}=1[/tex]what is a like term?
In an expression, two or more terms are like terms if they have the same variable and exponents.
For example the terms:
2a and -8a → these terms have the same variable "a" and the same exponent "1"
9y³ and 8x⁴ → these terms have different variables "y" and "x" and different exponents "3" and "4", so they are not like terms.
Constants, for example, -10 or 6, are also considered to be like terms.
If Carl wants to buy a $23,999 truck and put a 15% down payment on it, how much money should he save for a down payment?
Money Carl should save for a down payment is $3699.85.
A percentage is a number or ratio expressed as a fraction of 100. A percentage is a dimensionless number, it has no unit of measurement.
Calculation:-
Cost of truck = $23,999
Downpayment = 15% of truck price
So, downpayment value = $23,999 × 15/100
= $3699.85
To calculate the average percent, add all probabilities together as numbered values and divide by the sum of all the sets. Then multiply by using a hundred. The percentage may be calculated by dividing the price with the aid of the entire fee, after which multiplying the end result by a hundred. The method used to calculate the percent is: (value/general price)×100%.
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9. For each fraction, decimal, or percent, write the equivalent number from the list below 0.52, 38, 50, 0.35, 40% , 50 UN 76% 0.82 7 20 13 25
We have
In order to convert a fraction to a decimal, we divide the numerator between the denominator.
[tex]\frac{2}{5}=0.4=40\text{\%}[/tex][tex]0.82=\frac{82}{100}=\frac{41}{50}[/tex][tex]\frac{13}{25}=0.52[/tex][tex]76\text{\%}=\frac{76}{100}=\frac{38}{50}[/tex][tex]\frac{7}{20}=0.35[/tex]ANSWER
2/5=40%
0.82=41/50
13/25=0.52
76%=38/50
7/20=0.35
Select the correct answer from each drop-down menu.Wayne, Winston, and Wilfred walked for an hour. Winston and Wilfred walked the same number of miles. Winston walked 2 miles less than 2 themiles Wayne walked. Wilfred walked 2 miles more than 3 the miles Wayne walked.A variable selected to solve this problem should represent the number of mileswalked in an hour.In that hour, Wayne would have walkedmiles and Winston and Wilfred would have walkedmiles each. So, WaynewalkedWinston and Wilfred.
SOLUTION:
Winston =
[tex]Win\text{ston = }\frac{3}{2}\text{ (Wayne) - 2}[/tex][tex]\text{Wilfred = }\frac{1}{3\text{ }}\text{ wayne }+\text{ }\frac{3}{2}[/tex][tex]\frac{3}{2}\text{ wayne - 2 = }\frac{1}{3}\text{ wayne }+\frac{3}{2}[/tex][tex]\frac{3}{2}\text{ wayne - }\frac{1}{3}\text{ wayne = }\frac{3}{2}\text{ + 2}[/tex]Upon simplification, the number of miles wayne walked was 3
Substituting wayne = 3 into the second equation
[tex]\text{Wilfred = }\frac{1}{3\text{ }}\text{ (3) }+\text{ }\frac{3}{2}[/tex]Wilfred = 2.5
Since Wilfred and Winston walked the same number of miles,
Winston = 2.5
The first drop menu is Wilfred
The second drop menu is 3
The third drop menu is 2.5
The fourth drop menu is faster than.
Subtract 14 from 11 the difference is
First it is important to remember that, by definition, the result of a subtraction is called "Difference".
In this case you need to subtract 14 from 11. This can operation can be expressed as following:
[tex]11-14[/tex]Notice that , since 14 is greater than 11 and it is also a negative number, the sign of the result (the difference) must be negative too.
Therfefore, keeping the above on mind, you obtain that the difference is the following:
[tex]11-14=-3[/tex]Find the next number in the series
4, 8, 12, 20,-
●
32
34
36
38
-
Answer:
4, 8, 12, 20, 24, 28, 32, 34, 36, 38, 42, 46, 50...
Step-by-step explanation:
If it's counting by four, then the replacing number(s) are 24 and 28.
The next term of the sequence will be 32
What is the formula to calculate the nth term of an Arithmetic Sequence ?
The formula to calculate the nth term of an Arithmetic Sequence is -
a(n) = a + (n - 1)d
[a] - first term of A.P.
[d] - Common difference of A.P.
[n] - position of term
We have the following series -
4, 8, 12, 20 ...
We can write the [n]th term of this series as -
a[n] = a[n - 1] + a[n - 2]
So, for n = 5, we can write -
a[5] = a[4] + a[3] = 20 + 12 = 32
Hence, the next term of the sequence will be 32.
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what is the sum of a 7-term geometric series if the first term is -11, the last term is -45056, and the common ratio is -4
Answer:
[tex]-171,875[/tex]Explanation:
Here, we want to find the sum of the geometric series
Mathematically, we have the mathematical formula to calculate this as follows:
[tex]S_n\text{ = }\frac{a(1-r^n)}{1-r}[/tex]where:
a is the first term which is given as -11
n is the number of terms wich is 7
r is the common ratio which is -4
Substituting the values, we have it that:
[tex]\begin{gathered} S_n\text{ = }\frac{-11(1-(-4))\placeholder{⬚}^7}{1+4} \\ \\ S_n\text{ = }\frac{-11(5)\placeholder{⬚}^7}{5}\text{ = -11 }\times\text{ 5}^6\text{ = -171,875} \end{gathered}[/tex]Art club has 12 members. Each member paysmonthly dues of $12.60. On the first day of themonth, 4 members paid their dues. The remainingmembers paid their dues on the second day of themonth. How much money was collected in dues onthe second day of the month?
Given:
Total number of members in a club is 12
Each member pays $12.60 on every month.
[tex]\begin{gathered} \text{Number of members paid the dues on second day=12-4} \\ \text{Number of members paid the dues on second day=}8 \end{gathered}[/tex][tex]\begin{gathered} \text{Money collected on the second day=8}\times12.60 \\ \text{Money collected on the second day= \$100.80} \end{gathered}[/tex]Money collected on the second day of the month is $100.80
Answer:
100.8
Step-by-step explanation:
12-4 = 8
the 4 is the people who payes the first day the 8 is the people who payes the second day
12.60 eight times = 12.60•8= 100.8
the eight people each payes 12.60 so that would be 12.60 8 times
how we do this this is hoighs chbool clac 1 i failed it and i have to reatek it
The equation of the curve is given by:
[tex]y=5+\cot(x)-2\csc(x)[/tex]Differentiating both sides of the equation with respect to x, we have:
[tex]\frac{dy}{dx}=2\cot(x)\csc(x)-\csc^2(x)[/tex]Therefore, the slope of the tangent is given by the value of dy/dx when x= π / 2
[tex]2\cot(\frac{\pi}{2})\csc(\frac{\pi}{2})-\csc^2(\frac{\pi}{2})=-1[/tex]Using the point slope formula, it follows that:
[tex]\begin{gathered} y-3=-1(x-\frac{\pi}{2}) \\ y=-x+\frac{\pi}{2}+3 \end{gathered}[/tex]Therefore, the equation of the tangent at P is given by:
y = -x + π /2 + 3
Find the values that form the boundaries of the critical region for a two-tailed test with a = .05 for eachof the following sample sizes:a. n = 4b. n = 15c. n = 24
Given
a). n = 4
b). n = 15
c). n = 24
Find
values that form the boundaries of the critical region for a two-tailed test with a = .05
Explanation
a) n = 4
degree of freedom = n - 1 = 4 - 1 = 3
so , the t value for critical region =
[tex]\begin{gathered} \pm t_{0.05,3} \\ \pm3.182 \end{gathered}[/tex]b) n = 15
degree of freedom = 15 - 1 = 14
so , t- value =
[tex]\begin{gathered} \pm t_{0.05,15} \\ \pm2.131 \end{gathered}[/tex]c) n = 24
degree of freedom = 24 - 1 = 23
so , t - value =
[tex]\begin{gathered} \pm t_{0.05,23} \\ \pm2.069 \end{gathered}[/tex]Final Answer
Hence , the values that form the boundaries of the critical region for a two-tailed test with a = .05 are
a)
[tex]\pm3.182[/tex]b)
[tex]\pm2.131[/tex]c)
[tex]\pm2.069[/tex]A car that travels 35 mi in 40 min at a rate of…….miles per hour.
We want to determine the rate in miles per hour.
From the infomation given,
distance = 35 miles
time = 40 min
We would convert the time from minutes to hour. Recall,
60 minutes = 1 hour
40 minutes = x hour
By crossmultiplying,
60x = 40
x = 40/60 = 2/3 hour
This means that 40 minutes = 2/3 hour
Rate = distance/time
By substituting the values into the formula,
rate = 35/(2/3)
rate = 52.5
The 52.5 miles per hour
2x - 11 = -3
What does x equal?
Answer :x=4
Step-by-step explanation:
x equals a point. If you are in the same exact area but at a different x, you dont know how to get to the area where x is. It is important to note that x does not equal a point, but a location.
Solve the equation without using a calculator
[tex]x^2+\big(4x^3-3x\big)^2=1[/tex]
Answer:
[tex]x= \dfrac{\sqrt{2}}{2}, \quad x=-\dfrac{\sqrt{2}}{2},\\\\x=\dfrac{\sqrt{2 - \sqrt{2}}}{2}, \quad x=-\dfrac{\sqrt{2 - \sqrt{2}}}{2}, \quad x= \dfrac{\sqrt{2 + \sqrt{2}}}{2}, \quad x= -\dfrac{\sqrt{2 + \sqrt{2}}}{2}[/tex]
Step-by-step explanation:
Given equation:
[tex]x^2+(4x^3-3x)^2=1[/tex]
Expand and equal the equation to zero:
[tex]\begin{aligned}x^2+(4x^3-3x)^2&=1\\x^2+(4x^3-3x)(4x^3-3x)&=1\\x^2+16x^6-24x^4+9x^2&=1\\16x^6-24x^4+x^2+9x^2-1&=0\\16x^6-24x^4+10x^2-1&=0\end{aligned}[/tex]
Let u = x²:
[tex]\implies 16u^3-24u^2+10u-1=0[/tex]
Factor Theorem
If f(x) is a polynomial, and f(a) = 0, then (x – a) is a factor of f(x)
[tex]\textsf{As\;\;$f\left(\dfrac{1}{2}\right)=0$\;\;then\;$\left(u-\dfrac{1}{2}\right)$\;is a factor of $f(u)$}.[/tex]
Therefore:
[tex]\implies \left(u-\dfrac{1}{2}\right)\left(16u^2+bu+2\right)=0[/tex]
Compare the coefficients of u² to find b:
[tex]\implies b-8 = -24[/tex]
[tex]\implies b = -16[/tex]
Therefore:
[tex]\implies \left(u-\dfrac{1}{2}\right)\left(16u^2-16u+2\right)=0[/tex]
Factor out 2:
[tex]\implies 2\left(u-\dfrac{1}{2}\right)\left(8u^2-8u+1\right)=0[/tex]
[tex]\implies \left(u-\dfrac{1}{2}\right)\left(8u^2-8u+1\right)=0[/tex]
Zero Product Property
If a ⋅ b = 0 then either a = 0 or b = 0 (or both).
Using the Zero Product Property, set each factor equal to zero and solve for u.
[tex]\implies u-\dfrac{1}{2}=0 \implies u=\dfrac{1}{2}[/tex]
Use the quadratic formula to solve the quadratic:
[tex]\implies u=\dfrac{-(-8) \pm \sqrt{(-8)^2-4(8)(1)}}{2(8)}[/tex]
[tex]\implies u=\dfrac{8 \pm \sqrt{32}}{16}[/tex]
[tex]\implies u=\dfrac{8 \pm 4\sqrt{2}}{16}[/tex]
[tex]\implies u=\dfrac{2 \pm \sqrt{2}}{4}[/tex]
Therefore:
[tex]u=\dfrac{1}{2}, \quad u=\dfrac{2 - \sqrt{2}}{4}, \quad u=\dfrac{2 + \sqrt{2}}{4}[/tex]
Substitute back u = x²:
[tex]x^2=\dfrac{1}{2}, \quad x^2=\dfrac{2 - \sqrt{2}}{4}, \quad x^2=\dfrac{2 + \sqrt{2}}{4}[/tex]
Solve each case for x:
[tex]\implies x^2=\dfrac{1}{2}[/tex]
[tex]\implies x=\pm \sqrt{\dfrac{1}{2}}[/tex]
[tex]\implies x=\pm \dfrac{\sqrt{2}}{2}[/tex]
[tex]\implies x^2=\dfrac{2 - \sqrt{2}}{4}[/tex]
[tex]\implies x=\pm \sqrt{\dfrac{2 - \sqrt{2}}{4}}[/tex]
[tex]\implies x=\pm \dfrac{\sqrt{2 - \sqrt{2}}}{2}[/tex]
[tex]\implies x^2=\dfrac{2 + \sqrt{2}}{4}[/tex]
[tex]\implies x=\pm \sqrt{\dfrac{2 + \sqrt{2}}{4}}[/tex]
[tex]\implies x=\pm \dfrac{\sqrt{2 + \sqrt{2}}}{2}[/tex]
Solutions
[tex]x= \dfrac{\sqrt{2}}{2}, \quad x=-\dfrac{\sqrt{2}}{2},\\\\x=\dfrac{\sqrt{2 - \sqrt{2}}}{2}, \quad x=-\dfrac{\sqrt{2 - \sqrt{2}}}{2}, \quad x= \dfrac{\sqrt{2 + \sqrt{2}}}{2}, \quad x= -\dfrac{\sqrt{2 + \sqrt{2}}}{2}[/tex]
Answer the questions below.(a) Here are the prices (In thousands) for 10 houses for sale in a local neighborhood:$285, $286, $287, $290, $292, $295, $300, $301, $306, $307.which measure should be used to summarize the data?MeanMedianMode(b) in a survey, a soft drink company asks people to name as many brands of soft drinks as they can.Which measure glves the most frequently mentioned brand?MeanMedianMode(c) In the past 9 days, Kira has received the following numbers of email advertisements per day:40, 41, 43, 45, 48, 49, 50, 52, 85.Which measure should be used to summarize the data?O MeanMedianMode$2
a.
The data set shows the prices for houses.
Looking at the values, they lie near to the same value.
In this case, we can summarize the prices with the mean or median.
b. The survey was made to find how many names of brands of soft drinks they know. In this case, is important to know which soft drinks are the most popular.
Hence, the measure that gives the most frequently mentioned brand is the mode.
c. Kira has received many emails per day.
The emails also lie near to the same value except for the number of 85.
Where 85 represents an outlier (a value in a data set that is very different).
When we have outliers is better to use the median.