the current amount of employees today represents an increase of 26% from the previous year, this can be written as:
[tex]\begin{gathered} 1+26\% \\ 1+0.26 \\ 1.26 \end{gathered}[/tex]then,
[tex]44100=1.26x[/tex]where x is the number of employees the previous year, then, solve for x
[tex]\begin{gathered} x=\frac{44100}{1.26} \\ x=35000 \end{gathered}[/tex]Answer:
there were 35000 employees one year ago.
. Find the slope and y-intercept of the line shown below.to10-8--10-8-6-4-2.co6-4-2---2--4--6--8--10-yX4 6 8 10
Solution:
Given the graph:
Using two points on the line to find the slope, m:
[tex]\begin{gathered} (0,-1),(5,-2) \\ \\ x_1=0,y_1=-1,x_2=5,y_2=-2 \\ \\ m=\frac{y_2-y_1}{x_2-x_1} \end{gathered}[/tex][tex]\begin{gathered} m=\frac{-2-(-1)}{5-0} \\ \\ m=-\frac{1}{5} \end{gathered}[/tex]Thus, the y-intercept and slope, m of the line respectively are:
[tex][/tex]Describe the
transformation from the
graph of of to the graph
of g.
In the picture, there is table with functions. g(x)=-f(x)
The graph of g is negative 1 times of graph of f.
Given that,
In the picture, there is table with functions.
We have to find the transformation from the graph of f to the graph g.
The table has functions with the numbers.
x values as -2,-2,0,1
Function f(x) as 4,1,0,1
Function g(x) as -4,-1,0,-1
The f(x) function we can write as
f(x)=x²
And the g(x) function we can write as
g(x)=-x²
Therefore, g(x)=-f(x)
The graph of g is negative 1 times of graph of f.
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Express the trig ratios as fractions in simplest form.Drop down menu: (a) are equal (b) are not equal
ANSWER
[tex]\begin{gathered} \sin O=\frac{34}{35} \\ \\ \cos N=\frac{34}{35} \\ \\ \text{ They are equal.} \end{gathered}[/tex]EXPLANATION
We want to express the trigonometric ratios as fractions in the simplest terms.
The trigonometric ratios SOHCAHTOA for right triangles for sine and cosine are:
[tex]\begin{gathered} \sin\theta=\frac{opposite}{hypotenuse} \\ \\ \cos\theta=\frac{adjacent}{hypotenuse} \end{gathered}[/tex]Therefore, for the given triangle, we have that:
[tex]\begin{gathered} \sin O=\frac{34}{35} \\ \\ \cos N=\frac{34}{35} \end{gathered}[/tex]As we see from the equations above, the sine of O and the cosine of N are equal.
Omar and Zina Aboud found that the dealers cost of the base price was $16.558.16 and the dealer's options cost was $611.60. The consumer paid the $476.00 destination charge. If the percent of the dealer's cost is 92% and the percent of dealer's options cost is 88%, find the car's sticker price.
We want to calculate the sticker price. The sticker price is given by the formula
[tex]\text{sticker price = base }price+\text{ options + destination charge}[/tex]We are told that the destination charge is 476. We should determine the base price and the options to find the price sticker.
We are told that the dealer's cost of the base price is 92% of the pase price. So we have the equation
[tex]16558.16=\frac{92}{100}\cdot\text{base price}[/tex]so if we divide both sides by 92 and multiply by 100 we get
[tex]\text{base price = }16558.16\cdot\frac{100}{92}=17998[/tex]Now, applying the same principal for the options, we have
[tex]611.60=\frac{88}{100}\text{options}[/tex]which means that
[tex]\text{options}=611.6\cdot\frac{100}{88}=695[/tex]Replacing these values in the original equation we have that
[tex]\text{sticker price = }17998+695+476=19169[/tex]so the sticker price would be 19169
pls help this is due today and i have no idea what i am doing
(this isnt math but i didnt see music)
Why is it important for musicians to transcribe music?
Why do you think Beethoven’s story is unique (special)?
What are some ways that we can get better at active listening?
Below is a picture of a musical staff. What is a musical staff? What do the top lines and the bottom lines represent?
an image showing both staffs
What is a musical form? How does it make music more interesting?
Answer: look at explanation
Step-by-step explanation:
1. It helps you learn and listen to minor details in music better.
2. He had a handicap (partially deaf/deaf) but still managed to overcome and lay his mark in the music world.
3. Pay attention, eye contact, listen, pay attention to nonverbal cues, and show that you are listening
4. A foundation on which notes are drawn on. The lines represent a sound in music. A musical form is the structure of musical compositions. It makes music more interesting because it makes it organized and not jumbled up.
A sequence of positive integers with 2020 terms is called an FT sequence if each
term after the second is the sum of the previous two terms. For example, if the
first two terms of an FT sequence are 8 and 7, the sequence would begin
8,7,15, 22, 37,.... For some positive integer m, there are exactly 2415 FT sequences
where the first two terms are each less than 2m and the number of odd-valued
terms is more than twice the number of even-valued terms. What is the value of m?
(A) 21
(B) 69
(C) 115
(D) 35
(E) 105
The value of m will be 35.
Consider the FT 0, 1, 1, 2, 3, 5, 8, 13, 21, 24,45,.......
The numbers are arranged in order of even, odd, odd, even, odd, odd, even, odd, odd, even,.......
Hence, the loop contains 3 elements. If the number of terms is 2020 terms, then we have 673 loops + 1 element. (That is 3* 673 +1 = 2020) The last element will start the new loop and it is an even number.
In the other hand with 2019 terms, we have number of odd = 2* number of even. But the last term is even. That makes number of odd < twice number of even and it contradicts with the condition.
However in this case, the 2020th term is even and so there are fewer than twice as many odd valued terms as there are even-valued terms.
Thus, there are m2 + m × (m − 1) FT sequences that satisfy the required conditions.
Since there are 2415 such FT sequences, we may solve m2 + m × (m − 1) = 2415 by trial and error.
Evaluating m2 + m × (m − 1) when m = 30, we get 302 + 30 × 29 = 1770, and so m is greater
than 30.
When m = 33, we get 332 + 33 × 32 = 2145.
When m = 34, we get 342 + 34 × 33 = 2278.
When m = 35, we get 352 + 35 × 34 = 2415, as required.
Hence, Answer: (D)
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points on the coordinate plane below which point would lie on both the x-axis and the y-axis ___thats the question
We see that the unique point that lie on both the x-axis and the y-axis is the pont (0,0)!
find the coordinates of of the point at -270 degrees on circle radius 3.6 centered at the origin
Answer: [tex](0, 3.6)[/tex]
Step-by-step explanation:
When rotating -270 degrees, which is the same as 90 degrees, we are going to the topmost point of the graph.
Since the radius is 3.6, this point has coordinates [tex](0, 3.6)[/tex].
Sequence: 0, 4, 8, 12,...
Find the 18th term.
Answer:
68
Step-by-step explanation:
What is the slope of the line that passes through the points (4, -2) and (4, 10)?
Write your answer in simplest form.
The slope of the line that passes through the points (4, -2) and (4, 10) is 0. The slope of a line is 0 if it is horizontal. This function is constant.
What is slope?The ratio of the "vertical change" to the "horizontal change" between (any) two unique points on a line is used to compute slope. The ratio can also be written as a quotient ("rise over run"), which produces the same number for every two distinct points on the same line. A declining line has a negative "rise." The line might be useful, as determined by a road surveyor, or it might appear in a diagram that represents a road or a roof as a description or a design.
The slope's absolute value serves as a gauge for a line's steepness, incline, or grade. A steeper line is indicated by a slope with a higher absolute value. A line can be drawn with one of four directions: upward, downward, horizontal, or vertical.
If a line rises from left to right, it is said to be growing. The slope is upward, or m>0.
If a line slopes downward from left to right, it is diminishing. The slope, m0, is negative.
The slope of a line is 0 if it is horizontal. This function is constant.
A line's slope is ambiguous if it is vertical.
In other terms, the rise is (y2 y1) = y. The rise of a road between two points is the difference in the altitude of the road at those two sites, say y1 and y2. The run, or (x2 x1) = x, is the difference in distance from a given point measured along a level, horizontal line for relatively small distances where the Earth's curvature may be disregarded. Here, the ratio of the altitude change to the horizontal distance between any two places on the line is used to define the slope of the road between the two points.
In mathematical language, slope m of the line is
slope = [tex]\frac{x-x1}{y-y1} =\frac{4-4}{-2+10} = 0[/tex]
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Jared sells jewelry online. He can make 25 necklaces from 375 beads. Yesterday he made 21 necklaces. How many beads did Jared use to make necklaces yesterday
The number of beads he used to make 21 necklaces is 315 beads.
How to find the number of beads Jared use to make necklaces?He sells jewellery online. He can make 25 necklaces from 375 beads.
This means Jared uses 375 beads to make 25 necklaces.
Yesterday he made 21 necklaces.
Therefore, the number of beads he used to make the 21 jewellery can be calculated as follows:
Hence,
25 necklaces = 375 beads
21 necklaces = ?
cross multiply
number of beads used to make the necklaces = 21 × 375 / 25
number of beads used to make the necklaces = 7875 / 25
Therefore,
number of beads used to make the necklaces = 315
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A cosine function has an amplitude of 1/4 , period of pi/2, horizontal shift of 2pi, and vertical shift of -4.
What is the y-value of the positive function when x = 2π?
y = ?
The value of y when y = y = 0.25sin(4(x+2π))-4. and x = 2pi is -4.
Some functions (like Sine and Cosine) repeat forever and are called Periodic Functions.The Amplitude is the height from the center line to the peak (or to the trough).The Phase Shift is how far the function is shifted horizontally from the usual position.The Vertical Shift is how far the function is shifted vertically from the usual position.
We can have all of them in one equation:
y = A sin(B(x + C)) + D
amplitude is A
period is 2π/B
phase shift is C (positive is to the left)
vertical shift is D
So, for the given question we have amplitude is 0.25, period is 2π/4, phase shift is 2π, vertical shift is -4
So,
y = 0.25sin(4(x+2π))-4.
when x = 2π
y = 0.25sin(16π)-4
y = 0-4
y = - 4
Therefore, y = 0.25sin(4(x+2π))-4, and its y - value when x = 2π is -4.
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A pool fills up 3 gallons of water in one minute. How many pints does it fill in one minute?
In this problem we have to use the convertion factor that is equal to 1 so:
[tex]\frac{1gal}{8pn}=1[/tex]so we can made the transormation:
[tex]3\text{gal}\cdot\frac{8pn}{1\text{gal}}=24pn[/tex]So in one minute it fills 24 pints
A sample of 342 students at a university is surveyed. The students are classified according to gender ("female" or "male"). They are also classified according tomajor ("blology", "business", "engineering", "mathematics", or "computer science"). The results are given in the contingency table below.Biology Business Engineering Mathematics Computer scienceFemale4623501837Male4815461544What is the relative frequency of male students in the sample?Round your answer to two decimal places.0Х5.?
The relative frequency of an event is the quotient of the division between the event and the total number of the sample
From the given table
Add the numbers of males to find their total
[tex]48+15+46+15+44=168[/tex]Then add the numbers of females and males to find the total of the sample
[tex]168+46+23+50+18+37=342[/tex]Now, divide the number of males by the total to find the relative frequency
[tex]\begin{gathered} R\mathrm{}F=\frac{168}{342} \\ R\mathrm{}F=0.4912280702 \end{gathered}[/tex]Round it to 2 decimal places, then
The relative frequency of the male students = 0.49
Leonard is 68 years less than quadruple Sheldon's age. In 12 years, Leonard's age will be 24 years less than two times Sheldon's age. What are their 2 ages?
Sheldon is 16 4/7 years old, while Leonard is 21 1/7.
We are given the correlations between two numbers in this sort of query, specifically the data when their ages are contrasted now and in the future.
Let x represent Leonard's age.
We can infer Sheldon's age from the question as being 4x - 68.
The following equation results from the question:
x + 12 = 2((4x - 68) + 12) - 24
Fix x,
=> x + 12 = 2(4x - 68 + 12) - 24
x + 12 = 2(4x - 56) - 24
x + 12 = 8x - 112 - 24
x - 8x = -136 - 12
-7x = - 148
x = 148 / 7 or 21 1/7
Now, simply multiply 4x by 68 to get Sheldon's age.
=> 4x - 68
=> 4 * 148/7 - 68
=> 592/7 - 68
=> 116/7 or 16 4/7
Leonard is therefore 21 1/7 years old, while Sheldon is 16 4/7.
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2. The design of the Trieste was based on the design of a hot air balloon built by Auguste Piccard, Jacques's father. In 1932, Auguste rode in his hot-air balloon up to a record-breaking height.A. Auguste's ascent took 7 hours and went up 51,683 feet. Write a relationship y = kx to represent his ascent from his starting location.B. Auguste's descent took 3 hours and went down 52,940 feet. Write another relationship to represent his descent. C. Did Auguste Piccard end up at a greater or lesser altitude than his starting point? How much higher or lower?
ANSWER
A. y = 7383.3x
B. y = 17646.7x
C. Lesser altitude; 1257 feet
EXPLANATION
A. We want to write a proportional relationship between the time it took him to ascend and the total height he reached:
y = kx
where x = time taken
y = distance (height reached)
k = constant of proportionality
We have to find k for the relationship.
So, we have that:
[tex]\begin{gathered} 51683\text{ = k }\cdot\text{ }7 \\ \Rightarrow\text{ k = }\frac{51683}{7} \\ k\text{ = }7383.3\text{ ft/hr} \end{gathered}[/tex]This constant of proportionality (between distance and time) is called speed.
Therefore, the relationship for his ascent is:
y = 7383.3x
B. For his descent, we also have to find k in:
y = kx
We have that:
[tex]\begin{gathered} 52940\text{ = k }\cdot\text{ 3} \\ k\text{ = }\frac{52940}{3} \\ k=\text{ }17646.7\text{ ft/hr} \end{gathered}[/tex]Therefore, the relationship for his descent is:
y = 17646.7x
C. To find out if he reached a greater or lesser altitude than his starting point, we have to subtract his altitude during ascent from his altitude during descent.
If the value is positive then he ended up at a lesser altitude but if it is negative, he ended up on a greater altitude.
That is:
52940 - 51683
= 1257 feet
Since it is positive, that means that he descended more feet than he ascended and so, he ended up at a lesser altitude than his starting point, 1257 higher.
The US postal charges $.34 for the 1st ounce and $.23 for each additional ounce. use inequality to find the maximum number of whole ounces that can be mailed for $8.27
The charges for the first ounce = $0.34
Charges for each additional ounce = $0.23
Let the number of additional ounces be n
The maximum total charges = $8.27
The inequality that represents the given illustration is:
0.34 + 0.23n ≤ 8.27
0.23n ≤ 8.27 - 0.34
0.23n ≤ 7.93
n ≤ 7.93 / 0.23
n ≤ 34.48
Maximum number of whole ounces = Number of additional ouces + Initial ounce
Maximum number of whole ounces = 34.48 + 1
Maximum number of whole ou
Which function is the inverse of fix) = V5767O A.(-40) =3622 + 2. for = > 0O B.1-1(0) =652 + 2, for = ≥ 0O C.+ 2. for I > 0O D.f-11) = 6522. for r > 0
Given:
The function is
[tex]f(x)=\frac{\sqrt{x-2}}{6}[/tex]Required:
Find the inverse of the function.
Explanation:
The given function is:
[tex]f(x)=\frac{\sqrt{x-2}}{6}[/tex]Put
[tex]f(x)=y[/tex][tex]y=\frac{\sqrt{x-2}}{6}[/tex]Interchange x and y.
[tex]x=\frac{\sqrt{y-2}}{6}[/tex]Take the square on both sides.
[tex]x^2=\frac{y-2}{36}[/tex][tex]\begin{gathered} 36x^2=y-2 \\ y=36x^2+2 \end{gathered}[/tex]Substitute
[tex]y=f^{-1}(x)[/tex][tex]f^{-1}(x)=36x^2+2[/tex]Final Answer:
Option A is the correct answer.
A gardening shop receives a shipment of 12 crates of plants. Each crate contains 18 plants. A worker displays all the plants on 24 shelves with the same number of plants on each shelf. How many plants are displayed on each shelf?
A. 6
B. 16
C. 9
D. 36
Thanks!
Answer:16
Step-by-step explanation:
4to the fitth power times 4 negative 7 expnent
Answer:
-16,777,216
Step-by-step explanation:
4 to the fifth power =1024
-4 to the seventh power = 16, 384
1024 × -16, 384 =
- 16,777,216
Please help me with this question
Step-by-step explanation:
substitution means to use one equation to express one variable only in terms of the other variable and constants.
let's use
6f + 11b = 317
for that purpose.
6f = 317 - 11b
f = (317 - 11b)/6
now we use that my substituting "f" by its identical expression in b in the second equation :
7(317 - 11b)/6 + 5b = 221
to avoid handling fractions we multiply everything by 6 :
7(317 - 11b) + 30b = 1326
2219 - 77b + 30b = 1326
893 - 47b = 0
893 = 47b
b = 893/47 = 19 cents
f = (317 - 11b)/6 = (317 - 11×19)/6 = (317 - 209)/6 =
= 108/6 = 18 cents
a piece of fudge is 18 cents.
a piece of bubble gum is 19 cents.
The answer to which situation is best estimated by -7? How can you estimate decimal amounts?A The temperature falls 1.9°F from 4.3°F.B A hot air balloon rises 4.1 meters from the ground and then rises 2.3 meters more.C A submarine goes 3.8 feet below the surface of the ocean and then goes 2.5 feet farther downD The amount of money Sarah has after she buys milk for $3.85 and bread for $2.45.
Answer:
C A submarine goes 3.8 feet below the surface of the ocean and then goes 2.5 feet farther down
Explanation:
Option A
[tex]\begin{gathered} 4.3^0F\approx4^0F \\ 1.9^0F\approx2^0F \\ \text{Estimated change in temperature=}4^0F-2^0F=2^0F \end{gathered}[/tex]Option B
[tex]\begin{gathered} 4.1\text{ meters}\approx4\text{ meters} \\ 2.3\text{ meters}\approx2\text{ meters} \\ \text{Estimated change in the balloons's height}=4-2=2\text{ meters} \end{gathered}[/tex]Option C
[tex]\begin{gathered} -3.8f\exponentialE et\approx-4\text{ feet} \\ -2.5f\mathrm{e}et\approx-3\text{ feet} \\ \text{Estimated change in depth=}-4-3=-7\text{ feet} \end{gathered}[/tex]Option D
[tex]\begin{gathered} \$3.85\approx\$4 \\ \$2.45\approx\$2 \\ \text{Change in the amount Sarah has=-}\$4-\$2=-\$6 \end{gathered}[/tex]The situation that is best estimated by -7 is Option C.
11A family went out to dinner. The cost of their meal was $689, before sales tax and tip.A sales tax of 8% was added.(a)The family then tipped 20% on the amount after the sales tax was added,Part AWhat was the amount, rounded to the nearest cent, of just the sales tax?Show all of your work and record your final answer in the space below.Click the box below to type your math work and answer.On the right side of the box, click thebutton to start a new line.
The cost of the meal before the sales tax and tip= $689
Sales tax = 8%
The family then tipped 20% on the ammount after the sales was added
PART A
The amount of the sales tax can be calculated as follows
Amount of sales tax = sales tax * the amount of the meal
Amount of sales tax = 8% x $689
Amount of sales tax = 8/100 x 689
Amount of sales tax = 0.08 x 689
Amount of sales tax = $55.12
Therefore, the amount of sales tax is $55.12
PART B
What is the total amount paid by the family after adding the tax sales and 20% tipping
The cost of the meal = $689
Amount of sales tax = $55. 12
20% tipping on the amount after the sales tax was added
Amount of the meal after the sales has been added = $689 + $55.12
Amount of the meal after the sales tax has been added = $744. 12
20% tipping of the total amount = 20% x 744.12
= 20/100 x 744.12
= 0.2 x 744.12
= $148.82
The total amount paid by the family = 689 + 55.12 + 148.82
The total amount = $892.94
Therefore, the family paid a total amount of $892.94
Passes through the coordinates -2,2 parallel to the line whose slope is -1
The equation of the line is y-2x= 6.
Here the problem we are dealing with is related to the slope of the line, where, a slope of a line is the alter in the y coordinate about the alter in the x coordinate. The net change in the y-coordinate is spoken to by Δy and the net change in the x-coordinate is spoken to by Δx. The equation for the slope of a straight line is given by y − y1 = m(x − x1), where m is the slope and (x1,y1) are the points that pass through it,whereas the slope-intercept form of the line is given by y = mx + b, where b is the y-intercept. Since if two lines are parallel at that point it said the slope of both those lines are equal i.e m₁= m₂
Since it is given that the coordinates (-2,2) and the slope of the parallel to it, m=2
so the equation for the line is
=>y − y1 = m(x − x1)
=>y − 2 = 2(x +2)
=>y − 2 = 2x+4
=>y-2x= 6
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Find the eqaution of the line that passes through the coordinates -2,2 parallel to the line whose slope is -1.
Use substitution to solve the following system of equations. What is the value of y? {3x+2y=12{5x−y=7 A) y = -3B) y = 3C) y = -2D) y = 2
Answer
B) y = 3
Step-by-step explanation
Given the system of equations:
[tex]\begin{gathered} 3x+2y=12\text{ \lparen eq. 1\rparen} \\ 5x-y=7\text{ \lparen eq. 2\rparen} \end{gathered}[/tex]Isolating x from equation 1:
[tex]\begin{gathered} 3x+2y-2y=12-2y \\ 3x=12-2y \\ \frac{3x}{3}=\frac{12-2y}{3} \\ x=\frac{12}{3}-\frac{2}{3}y \\ x=4-\frac{2}{3}y\text{ \lparen eq. 3\rparen} \end{gathered}[/tex]Substituting equation 3 into equation 2 and solving for y:
[tex]\begin{gathered} 5(4-\frac{2}{3}y)-y=7 \\ 5\cdot4-5\cdot\frac{2}{3}y-y=7 \\ 20-\frac{10}{3}y-y=7 \\ 20-\frac{13}{3}y=7 \\ 20-\frac{13}{3}y-20=7-20 \\ -\frac{13}{3}y=-13 \\ (-\frac{3}{13})\cdot-\frac{13}{3}y=(-\frac{3}{13})\cdot-13 \\ y=3 \end{gathered}[/tex]
Answer:
y=3
Step-by-step explanation:
Got it right :/
Mathew deposits 400800.00 kina with the bank which offers 5½% interest per annum. calculate the interest earned after four years if it is compounded annually?
Solution
For this case we can use the following formula:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]where
A= future value, P= present value= 400800.00
r= 5.5%=0.055, n =1 , t= 4 years
n= represent the number of times that the interest is compounded each year =1 for this case
Replacing we got:
[tex]A=400800\cdot(1+\frac{0.055}{1})^{4\cdot1}=496520.92[/tex]then we can find the interest in the following way:
[tex]I=A-P=496520.92-400800=95720.919[/tex]Bryant measured a house and made a scale drawing the scale he used was 1 inch = 1 foot what scale factor does the drawing use?
The scale factor which is being used by this drawing is 1 : 12.
What is scale factor?A scale factor can be defined as the ratio of two (2) corresponding length of sides or diameter in two similar geometric figures such as equilateral triangles, river, planets in our solar system, etc.
Mathematically, the scale factor of a geometric figure can be calculated by using tis formula:
Scale factor = Dimension of image/Dimension of original figure
Generally speaking, an appropriate conversion factor to an equal value must be used when it is necessary to perform any mathematical conversion.
Conversion:
1 inch = 1 foot
12 inches = 1 foot
Therefore, the scale factor for this drawing is 1 : 12.
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A rectangle is 4 inches long and 2x inches wide. The value of the perimiter is equal to the value of the area
When a rectangle has a perimeter equal to the area and is 4 inches long and 2x inches wide, the value of x is 2.
What is the rectangle's perimeter?The sum of the lengths of a rectangle's four sides is referred to as its perimeter.A rectangle's perimeter is equal to 2(L + W).where W denotes the rectangle's width and L denotes its length.GIVEN:
The perimeter (in inches) of a rectangle with dimensions of 4 inches long by 2 inches wide equals the area (in square inches) specified in the question.Rectangle perimeter (P) equals 2x plus 2L.Rectangle area (A) equals LWAs per the stated circumstance,
P = A LW = 2W+2L 2x(4) = 2(2x)+2(4) (4) 8x = 4x +8 4x = 8 x=2HENCE,The value of x is 2 when a rectangle has a perimeter equal to the area, is 4 inches long, and is 2x inches broad.
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Please asnwer corrrectly and if you could explain or not
Answer:
y = 4·0.5^x
Step-by-step explanation:
If x is increased by 1, y is multiplied by 0.5.
x | y
0 | 4
1 | 4·0.5
2 | 4·0.5·0.5
3 | 4·0.5·0.5·0.5
State the number of possible real zeros and turning points of f(x) = x^7 – 6x^6 + 8x^5. Then determine all of the real zeros by factoring. 1) 7 real zeros and 6 turning points; 0, 2, and 42) 7 real zeros and 6 turning points; 0, –2, and –43) 6 real zeros and 5 turning points; 0, 2, and 44) 6 real zeros and 5 turning points; 0, –2, and –4
Step 1
The given equation is
[tex]f(x)=x^7-6x^6+8x^5[/tex]Required: To find the real zeroes by factorization and the turning points.
Step 2
Find the number of real zeroes
[tex]\begin{gathered} x^7-6x^6+8x^5=0 \\ x^5\mleft(x-2\mright)\mleft(x-4\mright)=0 \\ x^5=\text{ 0} \\ x\text{ = 0 five times} \\ ^{}or\text{ } \\ x-2\text{ = 0} \\ x=\text{ 2} \\ or \\ x-4=0 \\ x=\text{ 4} \\ \text{Therefore, there are 7 real zeroes} \end{gathered}[/tex]Step 3
Find the number of turning points.
[tex]f^{\prime}(x)\text{ = }7x^6-36x^5+40x^4[/tex][tex]\begin{gathered} As\text{ se}en\text{ from the differential, the number of turning points is found by subtracting 1 from the highest power of the function.} \\ \text{Hence, the number of turning points = 7-1 = 6} \\ \text{There are 6 turning points} \end{gathered}[/tex]Therefore, there are 7 real zeroes and 6 turning points; 0, 2 and 4
The answer is option A