The zeroes of the polynomial f(x)= x³ - 4x² + x + 6 = 0 are x - 1, x = 2
and x = 3.
We know that if x = a is one of the roots of a given polynomial x - a = 0 is a factor of the given polynomial.
To confirm if x - a = 0 is a factor of a polynomial we replace f(x) with f(a) and if the remainder is zero then it is confirmed that x - a = 0 is a factor.
Given, f(x)= x³ - 4x² + x + 6 = 0.
Now, zeroes of the polynomial should be factors of 6 they are ±1, ±2. ±3, ±6.
Now at x = 1 f(x) = 4 so not a zero, at x = - 1, f(x) = 0 so x = - 1 a zero
at x = 2 f(x) = 0 so x = 2 is a zero,
at x = 2 f(x) = so x = 3 is a zero.
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An investor invested a total of 57000 into two bonds. One bond paid 3% simple interest, and the other paid 2 7/8% interest. The investor earned a total of $1676.25 in annual interest. How much was originally invested in each account?
Answer:
$30,000 (at 3%) and $27,000 (at 2 7/8%)
Explanation:
The investor invested a total of 57000
• Let the amount invested at 3% simple interest = x
,• Then, the amount invested at 2 7/8% simple interest = 57000-x
Recall the formula for simple interest:
[tex]Simple\: Interest=\frac{Principal\times Rate\times Time}{100}[/tex]Interest at 3%
[tex]\begin{gathered} S\mathrm{}I\mathrm{}=\frac{x\times3\times1}{100} \\ =\frac{3x}{100} \\ =0.03x \end{gathered}[/tex]Interest at 2 7/8%
[tex]\begin{gathered} SI=\frac{(57000-x)\times2\frac{7}{8}\times1}{100} \\ =\frac{2.875(57000-x)}{100} \\ =0.02875(57000-x) \end{gathered}[/tex]Total Interest
The investor earned a total of $1676.25 in annual interest. Therefore:
[tex]\begin{gathered} 0.03x+0.02875(57000-x)=1676.25 \\ 0.03x+1638.75-0.02875x=1676.25 \\ 0.03x-0.02875x=1676.25-1638.75 \\ 0.00125x=37.5 \\ x=\frac{37.5}{0.00125} \\ x=30,000 \end{gathered}[/tex]Thus, the amount invested at 3% simple interest = $30,000
The amount invested at 2 7/8% simple interest = 57,000-30,000 = $27,000
the cone has a diameter of 10.5 inches and height of 6.75 inches. what is the volume of the cone? (hundredths place)the ball has a diameter of 8 1/2inches what Is the volume of the sphere? (hundredths place)the difference between the two volumes is(round to the nearest hundredth)
where,
r is the base radius of the cone
h is the perpendicular height of the cone
In this case,
r = 10.5 / 2= 5.25in
h = 6.75in
[tex]\begin{gathered} \text{Therefore} \\ \text{volume of cone}=\frac{\pi\times5.25^2\times6.75}{3}=194.83in^3 \end{gathered}[/tex]Volume of a ball is given by
[tex]\text{volume of ball =}\frac{4}{3}\times\pi\times r^3[/tex]where r is the radius of the ball
in this case,
[tex]r=\frac{8\frac{1}{2}}{2}=4.25in[/tex]Therefore
[tex]\text{Volume of ball = }\frac{4\times\pi\times4.25^3}{3}=321.56in^3[/tex][tex]\begin{gathered} \text{The difference betw}een\text{ the two volumes=321.56-194.83}=126.73 \\ =126.73in^3 \end{gathered}[/tex]A rectangular box, closed at the top, with a square base, is to have a volume of 4000 cm^ 3 . W What must be its dimensions (length, width, height ) if the box is to require the least possible material?
Solution
Area of square base of sides x is
[tex]Area=x^2[/tex]Volume = 4000cm^3
[tex]\begin{gathered} Volume=Bh \\ B=Base\text{ }Area \\ h=height \end{gathered}[/tex]Thus,
[tex]\begin{gathered} Volume=Bh \\ 4000=x^2h \\ \\ h=\frac{4000}{x^2} \end{gathered}[/tex]For the box to require the least possible material, is to simply minimize the surface area of the rectangular box
The surface Area is given as
[tex]\begin{gathered} Area=2(lw+wh+lh) \\ Since,\text{ it is a square base} \\ l=x \\ w=x \\ \\ Area=2(x^2+xh+xh) \\ Area=2(x^2+2xh) \\ Area=2(x^2+2x(\frac{4000}{x^2})) \\ \\ Area=\frac{16000}{x}+2x^2 \end{gathered}[/tex]Now, we differentiate
[tex]\begin{gathered} Area=\frac{16,000}{x}+2x^{2} \\ A=16000x^{-1}+2x^2 \\ By\text{ differentiating} \\ \frac{dA}{dx}=-16000x^{-2}+4x \\ \\ At\text{ minimum area, }\frac{dA}{dx}=0 \\ 4x=16000x^{-2} \\ x^3=4000 \\ x=10\sqrt[3]{4} \end{gathered}[/tex]Now, to find h
[tex]\begin{gathered} h=\frac{4000}{x^2} \\ h=\frac{4000}{100(4)^{\frac{2}{3}}} \\ h=4^{\frac{1}{3}}\times10 \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} Length=10\sqrt[3]{4}cm=15.874cm\text{ \lparen to three decimal places\rparen} \\ Width=10\sqrt[3]{4}cm=15.874cm\text{ \lparen to three decimal places\rparen} \\ height=15.874cm\text{ \lparen to three decimal places\rparen} \end{gathered}[/tex]UVW is a regular triangle and the length of UV is 3 meters. What is the length of UW?A) 2 mB) 3 mC) 6 mD) 9 m
When a polygon is regular, all its sides are congruent (that is, they have the same length).
So, if the triangle is regular and one of the sides has a length of 3 meters, therefore all sides have a length of 3 meters.
Correct option: B.
Hope earns $12.35 an hour plus time and a half for weekend work. Last week she worked her regular 45 hours plus 16 hours of overtime on the weekend.What was her total pay for the week?
Answer:
$852.23
Explanation:
Given the hourly rate as $12.35, let's go ahead and determine what Hope earns per hour for weekend work as seen below;
Time and a half rate (Overtime rate) = 1.5 x 12.35 = $18.53
So if Hope worked her regular 45 hours plus 16 hours of overtime on the weekend, we can go ahead and determine her total pay as seen below;
Total pay = Regular wages + Overtime Wages
= (12.35 x 45) + (18.53 x 16)
= 555.75 + 296.48
= $852.23
Writing the equation for each line. slope 6 and y-intercept (0,-2).
We are given slope and the y-intercept. The formula y = mx + c can be used to determine the equation of the line.
m = slope
c= y-intercept.
For this question,
m = 6
c = -2
The equation of the line is
y = 6x + (-2)
y = 6x -2
The answer is y = 6x -2
A flower garden is shaped like a circle. Its radius is 18 yd. A ring-shaped path goes around the garden. The width of the path is 5 yd.
The gardener is going to cover the path with sand. If one bag of sand can cover 6 yd^2, how many bags of sand does the gardener need? Note that sand comes only by the bag, so the number of bags must be a whole number. (Use the value 3.14 for pi.)
The number of gallons of the coating must be a whole number will be 114 gallons.
What is the area of the circle?Let d be the diameter of the circle. Then the area of the circle will be
A = (π/4) d² square units
At the recreation area, there is a pool molded like a circle with a measurement of 24 yds. A ring-molded way circumvents the pool. Its width is 6 yds.
The area of the path will be given as,
A = (π/4) (24 + 6 + 6)² - (π/4) (24)²
A = (π/4) [36² - 24²]
A = (π/4) [720]
A = 565.5 square yards
We will give another layer of covering to the way. On the off chance that one gallon of covering can cover 5 yd². Then the number of gallons of coating is given as,
⇒ (565.5) / 5
⇒ 113.097
⇒ 114 gallons
The number of gallons must be a whole number will be 114 gallons.
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VFind the area of the figure. (Sides meet at right angles.)3 in5 in5 in10 in8 in
Given the shown composite figure
We will find the area of the figure using the following figure
as shown, the figure is divided into 2 shapes
shape (1) is a rectangle with dimensions 3 in and 5 in
The area of shape (1) = 3 x 5 = 15 in²
shape (2) is a rectangle with dimensions 8 in and 5 in
The area of shape (2) = 8 x 5 = 40 in²
The total area of the figure = 15 + 40 = 55 in²
So, the answer will be Area = 55 in²
can you please solve this practice problem for me I really need assistance and also ONLY SOLVE THE SECOND PROBLEM.
The answer is the third choice
x - 5 because the temperature diminishes 5 degrees
2 because the temperature doubled
40 because the final number is 40
After a dilation with center (0,0), the image of DB is D'B'. IDB = 4.5 and D'B' = 18, the scale factor of this dilation is (1) (3) (2) 5 (4) 4
We can write the length of DB like this:
[tex]4.5=\frac{9}{2}[/tex]Now, to find the scale factor of the dilation, we have to solve for k the following equation:
[tex]\frac{9}{2}k=18[/tex]then, we have the following:
[tex]\begin{gathered} \frac{9}{2}k=18 \\ \Rightarrow9k=18\cdot2=36 \\ \Rightarrow k=\frac{36}{9}=4 \\ k=4 \end{gathered}[/tex]therefore, the scale factor is k=4
(1 4/7)_____-(15/21)
(1 4/7)
_____
-(15/21)
To solve;
we will cange the division to multiplication. By doing that 15/21 becomes 21/15
That is;
[tex]\frac{14}{7}\times-\frac{21}{15}[/tex]7 can divide 21 to give 3
Hence;
[tex]\frac{14}{1}\times-\frac{3}{15}[/tex]3 can also divide 15 to give 5
[tex]\frac{14}{1}\times-\frac{1}{5}[/tex]we will now multiply
[tex]=\frac{-14}{5}[/tex][tex]=-2\frac{4}{5}[/tex]the probability of the complement of an event is _______ less than the probability of the event itd self possible answers sometimesalwaysnevernot enough information provided to answer the question
The complement rule states that the sum of the probabilities of an event and its complement must equal to 1. That is, for an event A and its complement A', we have
[tex]P(A)+P(A^{\prime})=1[/tex]so, as long as the sum is 1, the probability of the complement of an event is sometimes less than the probability of the event itself .
Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value.n=34 and 4i are zeros;f(-1)=85F(x) = ________
Given that at n=3: 4 and 4i are zeros:
Then:
[tex]\begin{gathered} (x-4)(x-4i)(x+4i)\Rightarrow(x-4)(x^2-(4i)^2) \\ (x-4)(x-4i)(x+4i)\Rightarrow(x-4)(x^2+16) \\ (x-4)(x-4i)(x+4i)\Rightarrow x^3+16x-4x^2-64 \\ (x-4)(x-4i)(x+4i)\Rightarrow x^3-4x^2+16x-64 \end{gathered}[/tex]Hence the function is:
[tex]F=x^3-4x^2+16x-64[/tex]A gardener wishes to create raised beds. She wants the bedsto follow the golden ratio. If the shorter side is 4 feet, whatwill be the longer side of the beds? Round answer to the nearesttenth.
The golden ratio is equal to 1.618. This means that the ratio between the two sides of the bed must be equal to that value. We were given the shorter side of the bed, therefore to find the length of the longer side we need to multiply the short one by the golden ratio.
[tex]\text{longer = 1.618}\cdot4=6.472\text{ ft}[/tex]The longer side will be approximately 6.5 ft
AT"U"V is the translation of AT"U"V. what is the translation rule? ("A" is actually a triangle)
we know that
The coordinate of point U is (-7,8) see the image
The coordinate of point U' is (7,0) see the image
so
The rule of the trnslation is 14 units to the right and 8 units down
therefore
the rule is
(x,y) ------> (x+14,y-8)Find the slope of each line
The equation of the slope of a line is given by the formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]The coordinates (x1, y1) and (x2, y2) are the coordinates that we need to identify in the graph:
(x1, y1) = (-4, 4)
(x2, y2) = (0, -3)
Then, applying the formula to find m:
[tex]m=\frac{y2-y1}{x2-x1}=\frac{-3-4}{0-(-4)}=\frac{-7}{4}\rightarrow m=-\frac{7}{4}[/tex]Therefore, the slope for the line is m = -7/4.
This is the last question we have been stuck on it is there anyway you can help us out
Clearly,
[tex]1.3\cdot10^{12}[/tex]Is the largest number.
That's because it has more zeros due to its exponent is the most bigger of the four options.
Seth's father is thinking of buying his son a summer movie pass for $30 dollars. With the pass, matinees cost $2 each. Without thepass each movie is normally $6. If Seth plans to go to 30 movies this summer how much money would he save?
Theres a number of passes to purchase
Normal price (without pass) is $6
If Seth goes to all 30 movies without pass he will spend
30x6 = $180 dollars
WITH the pass Seth have $30 dollars , then dividing by 2 , this means he can go to 30/2 = 15 movies in summer with pass
Then remains another 15 movies that he can vo WITHOUT PASS
in this 15 movies Seth spends 15x6= $90 dollars
add this to the $30 spent in the other 15 movie
it gives 30+90 = $120 dollars
Finally substract
$180- $120= $60 dollars he can save
Which Story can be represented as 36 divided by 4 A. Jim has 36 pennies. May has four more pennies Than Jin.B. Jin has 4 fewer pennies than May. May has 36 pennies. C.Jin has 4 pennies. May has 36 times as many pennies as Jin.D.Jin has 36 pennies. She shares the pennies equally among 4 friends.
D is the answer.
In all the other cases the operations that we will have to make are not divisions.
In case A we have an addition.
In case B we have a substraction.
In case C we have a multiplication.
The correct story which an be represented as 36 divided by 4 is,
D. Jin has 36 pennies. She shares the pennies equally among 4 friends.
We have to given that,
All expressions are,
A. Jim has 36 pennies. May has four more pennies Than Jin.
B. Jin has 4 fewer pennies than May. May has 36 pennies.
C. Jin has 4 pennies. May has 36 times as many pennies as Jin.
D. Jin has 36 pennies. She shares the pennies equally among 4 friends.
Hence, In all the other cases the operations that we will have to make are not divisions.
In case A we have an addition.
In case B we have a subtraction.
In case C we have a multiplication.
In case of D;
D. Jin has 36 pennies. She shares the pennies equally among 4 friends.
This shows the operation division.
As, Each friends get, 36 / 4 = 9 pennies
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What is the measure of angle B in this triangle?
Enter your answer in the box.
Answer:
Step-by-step explanation:
First, you add the degrees available to you, next, you need to count the variable and add them together, in this case, your problem should show, 40+20-30 for your degrees, which is 30 degrees. Next, you have an "x" and a "2x," therefore you add them together to get 3x. After doing this, you get to your 2-step equation, which should look like 30 + 3x = 180. You get 180 because that is the length of the triangle, it should always be 180. You can now start by subtracting 180 - 30, we switch the + and - when doing the first part. Now, you should have gotten, 150, with 150 you can divide it by 3, you divided by 3 because you switch the division and the multiplication, now you do 150 divided by 3, which is 50. YOUR ANSWER IS: x = 50.
The measures of the angles of a triangle are shown in the figure below. Solve for x. 47 74
ThisThe sum of angles in a triangle
[tex]=180^0[/tex]The angles given in the triangle are
[tex]x^0,74,47^0^{}^{}^{}_{}^{}[/tex]Which means that,
[tex]x^0+47^0+74^0=180^0[/tex][tex]\begin{gathered} x^0+121^0=180^0 \\ \text{collecting like terms,we will have} \\ x^0=180^0-121^0 \\ x^0=59^0 \end{gathered}[/tex]Hence,
The final answer is x=59°
The following picture has the question that has me troubled can you show me how to get to the answer
Given that:
[tex]f(x)=\sqrt[]{x},x_0=8,dx=0.07[/tex]Find the change.
[tex]\begin{gathered} \nabla f=f(8+\text{0}.07)-f(8) \\ =f(8.07)-f(8) \\ =\sqrt[]{8.07}-\sqrt[]{8} \\ =0.0123474175 \end{gathered}[/tex]Find the estimate value.
[tex]\begin{gathered} df=\frac{1}{2\sqrt[]{8}}\cdot0.07 \\ =0.0123743687 \end{gathered}[/tex]Determine the approximate error.
[tex]\begin{gathered} |\nabla f-df|=|0.0123474175-0.0123743687| \\ =0.0000269512 \\ \approx0.00002 \end{gathered}[/tex]Option B is correct.
Please get help with this for I have tried many times to get the correct answers for each but still could not
A dilation of a point with a factor k is given by:
[tex](x,y)\rightarrow(kx,ky)[/tex]In this case the dilation factor is ; which means that we need to multiply each component of the vertexes by two. After doing this we get the following graph for the original and final polygon:
From the figure we can answer the questions.
a)
Longest side length of the original figrue: 3 units
Longest side lenght of the final figure: 6 units.
b)
Longest side length of the final figure = 2 x Longest side length of the original figure.
c)
For a positive scale final figure is always bigger than the original one, therefore, the stament is False.
d)
Any dilation creates a similar figure since we are not changing the angles, only the lengths. Therefore the statement is True
wХ14. Given: WX || YZ, WX = YZProve: AWXZ AYZXZ
The figure is rotated 360° clockwise with the origin as the center of rotation. Which graph represents the rotated figure?
Given
The figure is rotated 360° clockwise with the origin as the center of rotation. Which graph represents the rotated figure?
Answer
After the rotation of 360 degrees, a figure comes back to original position
Option A is correct
11. Which set of points could you use to create a line with slope of -3/2? (A) (5,7) (7,4) (B) (-1,4) (1,7) (C) (3,2) (1,-3)(D) (-3,0) (0,-2)
Answer:
(A) (5,7) (7,4)
Explanation:
To determine the set of points that could be used to create a line with a slope of -3/2, we use the slope formula below.
[tex]\text{Slope}=\frac{Change\text{ in y-axis}}{Change\text{ in x-axis}}[/tex]Option A
[tex]\begin{gathered} \text{Slope}=\frac{7-4}{5-7} \\ =-\frac{3}{2} \end{gathered}[/tex]Therefore, the set of points is (5,7) and (7,4).
you own 5 pairs of jeans and want to take 2 of them on vacation with you. in how many ways can you choose 2 pairs of jeans
SOLUTION:
Step 1:
In the question, we are given the following:
you own 5 pairs of jeans and want to take 2 of them on vacation with you.
In how many ways can you choose 2 pairs of jeans?
Step 2:
The details of the solution are as follows:
From this question, we can see clearly that this is an application of selection under combinatorial analysis:
[tex]n\text{ C}_r=\text{ }\frac{n!}{r!(\text{ n - r \rparen}!}[/tex][tex]\begin{gathered} Now\text{, we have that:} \\ \text{n = 5} \\ \text{r = 2.} \\ Then,\text{ we have that:} \\ 5\text{ C}_2\text{ = }\frac{5!}{2!\text{ \lparen 5- 2\rparen}!}=\text{ }\frac{5!}{2!3!}=\frac{5\text{ x 4 x 3}!}{2!\text{ x 3}!}=\frac{20}{2}=\text{ 10 ways} \end{gathered}[/tex]CONCLUSION:
The final answer is:
[tex]10\text{ ways}[/tex]
One gram is approximately 2.2 x 10-pounds. Which of the following represents this number in standard notation? OA. 0.0022 OB. 0.022 OC. 2200 OD. 0.00022
One gram is approximately 2.2 x 10^-3
10^-3 = 1/1000
= 0.001
= 2.2 x 0.001
= 0.0022
The answer is option A
Complete the table for the missing values of r .PT101812161416
To complete the table, it is observed that as p-values increase by 2 downward, the r-value decreases by 2
Thus
p r
10 18
12 16
14 14
16 12
The equation for the number of points Sydney's needed is
r = 28 - p
6×(11_6)÷2= do get this
Ayden, remember the order of operations PEMDAS.
1. Solve the Parentheses First
2. Solve the Exponents
3. Solve the Multiplication and Division
4. Finally, solve the Addition and Subtraction
Solving our exercise, we have:
6× (11 - 6) ÷ 2 =
6 x 5 / 2 =
30/2 =
Congratulations, Ayden! You did it!