SOLUTION
We want to write
[tex]\begin{gathered} \mleft(3-2i\mright)^3\text{ in simplest form } \\ a+bi \end{gathered}[/tex]This means we have to expand
[tex](3-2i)^3[/tex]Applying perfect cube formula, we have
[tex]\begin{gathered} \mleft(a-b\mright)^3=a^3-3a^2b+3ab^2-b^3 \\ \text{where } \\ a=3,\: \: b=2i \end{gathered}[/tex]We have
[tex]\begin{gathered} (a-b)^3=a^3-3a^2b+3ab^2-b^3 \\ \mleft(3-2i\mright)^3=3^3-(3\times3^2\times2i)+(3\times3\times(2i)^2)-(2i)^3_{} \\ =27-(27\times2i)+(9\times(2i)^2)-(2i)^3_{} \end{gathered}[/tex]This becomes
[tex]\begin{gathered} \text{note that i = }\sqrt[]{-1} \\ i^2=\sqrt[]{-1^2}=-1 \\ So\text{ we have } \\ =27-(27\times2i)+(9\times(2i)^2)-(2i)^3_{} \\ 27-54i+(9\times4i^2)-(8i^2\times i) \\ 27-54i+(9\times4\times-1)-(8\times-1\times i) \\ 27-54i-36+8i \\ -9-46i \end{gathered}[/tex]Hence the answer is
[tex]-9-46i[/tex]Which expressions have the fewest significant Figures?A. 18.8 - 6.5B. 4350 - 2210C. 15.4 - 8.1D. 54.5 * 30.7
Answer
Option C is obviously the answer.
Explanation
It will be easy to answer this by providing the answers to the expressions.
Option A
18.8 - 6.5 = 12.3 (3 significant figures)
Option B
4350 - 2210 = 2140 (4 significant figures)
Option C
15.4 - 8.1 = 7.3 (2 significant figures)
Option D
54.5 * 30.7 = 1673.15 (6 significant figures)
Hope this Helps!!!
To prepare an aquarium for use, you can clean it with saltwater solution.The amount of salt varies directly with the volume of the water.The solution has 3 teaspoons of aquarium salt for every 2 gallons of water.
teaspoons of water = y
gallons of water = x
• a)
y = k x
Where k is the constant of proportionality.
Replace x,y by the values given and solve for k:
3= k 2
3/2 = k
k= 1.5
Equation:
y= 1.5x
• b) replace x=10 and solve for y
y= 1.5x
y= 1.5 (10)
y= 15
15 teaspoons of aquarium salt
• c) replace y= 39 and solve for x
y=1.5x
39 = 1.5 x
39/1.5= x
x = 26
26 gallons of water
Given (12 ,7)and (X,-8) find all x such that the distance between these two points is 17 separate multiple answers with a comma
X=20
Explanationthe distance between 2 points is given by
[tex]\begin{gathered} for \\ P1(x_1,y_1) \\ P2(x_2y_2) \\ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \end{gathered}[/tex]so
Step 1
a)given
[tex]\begin{gathered} P1(12,7) \\ P2(x,-8) \\ d=17 \end{gathered}[/tex]b) now, replace in the formula and solve for x
[tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ 17=\sqrt{(x-12)^2+(-8-7)^2} \\ 17=\sqrt{(x-12)^2+(-15)^2} \\ raise\text{ both sides to power 2} \\ 17^2=(\sqrt{(x-12)^2+(-15)^2})^2 \\ 289=(x-12)^2+225 \\ subtract\text{ 225 in both sides} \\ 289-225=(x-12)^2+225-225 \\ 64=(x-12)^2 \\ square\text{ root in both sides} \\ \sqrt{64}=\sqrt{(x-12)^2} \\ 8=x-12 \\ add\text{ 12 in both sides} \\ 8=x-12 \\ 8+12=x-12+12 \\ 20=x \end{gathered}[/tex]therefore, the answer is
X=20
I hope this helps you
Below is the graph of a trigonometric function. It intersects its midline at (-1.7, -10) and again at(5.1, -10). What is the period
you can get the Period graphically if you see the points A and B are the upper points of the graph. if you subtract them you get the period
in this case, let me see
Here they give you the points where the graph cross the midline, if you see, that is half of the period, so
Period= 2* (5,1-(-1,7))= 13,6
You have to subtract the points to know the distance between that points, and the graph is
when you subtract the points, you are getting the distance between (for example) the green point on my draw
And the period is the time it takes for one complete oscillation, after this, it repeats over and over
So between 3 of these green points we have a period
to know the value, we need to know the distance between them (in this case (5,1-(-1,7))= 6,8
each square have 1 period on it, and the value is 2 times the distance between the green points
Answer:
Period= 2* (5,1-(-1,7))= 13,6
The circumference of a circle is 37.68 meters, what is the radius?
A.Find the equation of the line of best fit round one decimal place, if needed choose the correct answer
B.the correlation coefficient is
C. The predicted number of cars sold in year 10 is
An equation for the line of best fit is equal to: D. y = 4.8x + 10.
The correlation coefficient, R² is equal to 0.988.
The predicted number of cars sold in year 10 is equal to 490 cars.
How to find an equation of the line of best fit for the data?In order to determine a linear equation for the line of best fit (trend line) that models the data points contained in the table, we would have to use a graphing calculator (scatter plot).
In this scenario, the number of years would be plotted on the x-axis of the scatter plot while the number of cars sold would be plotted on the y-axis of the scatter plot.
On the Excel worksheet, you should right click on any data point on the scatter plot, select format trend line, and then tick the box to display an equation for the line of best fit (trend line) on the scatter plot.
From the scatter plot (see attachment) which models the relationship between the number of years and cars sold in the table, a linear equation for the line of best fit is given by:
y = 4.8x + 10
Next, we would determine the predicted number of cars sold in year 10:
y = 4.8x + 10
y = 4.8(10) + 10
y = 480 + 10
y = 490 cars.
Read more on scatter plot here: brainly.com/question/28605735
#SPJ1
I’m the diagram below, C is the midpoint of AB. If AC is 4 centimeters, what is the length of CBA.4cmB.8cmC.6cmD.2cm
Consider that a mid-point is at equal distances from each end of the line segment.
Given that C is the mid point of AB, so point C must be at equal distance from ends A and B,
[tex]AC=CB[/tex]Given that AC measures 4 centimeters,
[tex]AC=4\text{ cm}[/tex]Substitute the value,
[tex]CB=4\text{ cm}[/tex]Therefore, option A is the correct choice.
Find the value of the logarithmic expression. log subscript 6 216log subscript 6 216=____?
Solution
For this case we have the following:
[tex]\log _6(216)=3[/tex]The reason is because:
[tex]6^3=6\cdot6\cdot6=216[/tex]what's the answer please help
Domain are the "x"'s in this example Domain = [-2, 2]
Range are the "y" in this example Range = [-3, 3]
The second choice is correct.
What is the solution to the following equation? X +(-13) = -5 A. X= 18 B. X = 8 C. x -18 D. X = -8
Given the following expression:
X + (-13) = -5
Let's determine the value of X.
X + (-13) = -5
X - 13 = -5
X = -5 + 13
X = 8
Therefore, X = 8.
Simplify or expand each expression and then Classify it by its degree and number of terms
2x^3 (7x^2 + 3x + 1) - 4x^4
multiply 2x^3 in
14x^5 + 6x^4 + 2x^3 - 4x^4
14x^5 + 2x^4 + 2x^3
3 terms, the degree is 5
terms are single expressions such as (2x^3)
the degree is the highest exponent
(6x - 2x^4 + 5x^6) + (7x^4 - 3x^6 + 9)
combine
6x - 2x^4 + 5x^6 + 7x^4 - 3x^6 + 9
simplify
2x^6 + 5x^4 + 6x + 9
4 terms, the degree is 6
(x^2 + 9)(x^2 - 9)
FOIL, (a + b)(c + d) -> ac + ad + bc + bd
x^2 * x^2 + x^2 * -9 + 9 * x^2 + 9 * -9
simplify
x^4 - 81
2 terms, the degree is 4
how long will it take for the population to get to 2552 alligators?
we have the equation
[tex]P(t)=319(2)^{(\frac{t}{3})}[/tex]For P(t)=2,552
substitute in the given equation
[tex]2,552=319(2)^{(\frac{t}{3})}[/tex]solve for t
[tex]\begin{gathered} 2,552=319(2)^{(\frac{t}{3})} \\ \frac{2,552}{319}=(2)^{(\frac{t}{3})} \end{gathered}[/tex]Apply log both sides
[tex]\begin{gathered} \log \lbrack\frac{2,552}{319}\rbrack=\log \lbrack(2)^{(\frac{t}{3})}\rbrack \\ \log \lbrack\frac{2,552}{319}\rbrack=\frac{t}{3}\cdot\log (2) \end{gathered}[/tex]t=9 yearsthe answer is 9 years from the time of introductionThe figure below is a net for a right rectangular prism. Its surface area is 396 cm2 andthe area of some of the faces are filled in below. Find the area of the missing faces,and the missing dimension.
hello
to solve this question, let's add up all the areas from the sides given and equate it to the total area of the prism. Then we can also denote the side with the missing area as x
[tex]\begin{gathered} 396=42+72+42+72+x+x \\ 396=228+2x \\ 2x=396-228 \\ 2x=168 \\ \text{divide both sides by the coefficient of x} \\ \frac{2x}{2}=\frac{168}{2} \\ x=84 \end{gathered}[/tex]now we have established the area of the missing sides as 84cm^2
but then from careful observation, the figure with the missing side have a shape of a rectangle and we can use the formula of area of a rectangle to find the missing side.
[tex]\begin{gathered} a=l\times w \\ a=84\operatorname{cm}^2 \\ l=? \\ w=7\operatorname{cm} \\ 84=l\times7 \\ 84=7l \\ \frac{84}{7}=\frac{7l}{7} \\ l=12\operatorname{cm} \end{gathered}[/tex]from the calculations above, the missing side is equal to 12cm
What is the equation in standard form of the line that passes through the point (6,-1) and isparallel to the line represented by 8x + 3y=15?A 8x+3y=-45B 8x-3y = -51C 8x+3y=45D 8x - 3y=51
The slope of the given line is:
[tex]s=-\frac{8}{3}\text{.}[/tex]Therefore, the slope of a parallel line to the given line must be -8/3.
Using the slope-point formula for the equation of a line we get:
[tex]y-(-1)=-\frac{8}{3}(x-6)\text{.}[/tex]Taking the above equation to its standard form we get:
[tex]\begin{gathered} y+1=-\frac{8}{3}(x-6), \\ 3y+3=-8(x-6), \\ 3y+3=-8x+48, \\ 8x+3y=48-3, \\ 8x+3y=45. \end{gathered}[/tex]Answer: Option C.
Peter is thinking of a number. If he adds 23 to that number, the sum is 31.A. Write an algebraic equation you can use to find Peter’s number. Let n be Peter’s number.
The length of a rectangle is 1 inch shorter than twice the width (x).Which is the width (x) when the area (y) = 3321 square inches?
the width (x) of the rectangle = 41 inches
Explanation:
let the width = x
twice the width = 2x
The length of a rectangle is 1 inch shorter than twice the width (x) = 2x - 1
length = 2x -1
Area of rectangle = length × breadth
area (y) = 3321 square inches
y = x × 2x - 1 = x(2x - 1)
3321 = 2x² - x
2x² - x - 3321 = 0
We use factorisation to find x:
a = 2, b = -1, c = -3321
a × c = 2(-3321) = -6642
The factors which gives -1 when we sum together but gives -6642 when we multiply together are -82 and +81
2x² -82x + 81x - 3321 = 0
2x(x - 41) + 81(x - 41) = 0
(2x + 81) (x - 41) = 0
(2x + 81) = 0 or (x - 41) = 0
2x + 81 = 0
2x = -81
x = -81/2 inches
(x - 41) = 0
x - 41 = 0
x = 41 inches
Since we can't have a negative number as the width, the width (x) of the rectangle = 41 inches
Destiny needs a new coat for the winter and she found one at Old Navy for $48.88. The sale tax is 8%. How much will she pay for the sales tax? How much will she pay all together? SHOW ALL OF YOUR WORK.
Hello!
First, let's consider the value of $48.88 as 100%. Then, we have to find how many correspond to 8% of it. We can calculate it using the rule of three:
[tex]\begin{gathered} \frac{48.88}{x}=\frac{100}{8} \\ \\ \text{ multiplying across} \\ x\cdot100=8\cdot48.88 \\ 100x=391.04 \\ x=\frac{391.04}{100} \\ x\cong3.91 \end{gathered}[/tex]So, the sales tax is $3.91.
Now, we have to add the price of the coat with the sale tax:
$48.88 + $3.91 = $52.79
If she pays all together, the cost will be $52.79.
wich of the following is the correct value of 0.22 0.4?
Answer:
The decimal place will be placed to the left by the total number of digits after the two numbers.
Explanation:
The multiplication we are asked to perform is
[tex]1.35\times4.2[/tex]which has a total of 3 digits after the decimal point.
Now we know that
[tex]135\times42=5670[/tex]therefore, to find 1.35 x 4.2 we just shift the decimal place in the above to the left by 3 units
therefore
[tex]1.35\times4.2=5.67[/tex]which is our answer!
Michael says that 5 = 5 Is his answer correct? Explain. (1 O A. 00:00 Yes. Both expressions are equal to 625 СВ. 00:00 Yes Both expressions are equal to O C. 00:00 No. The first expression is equal to and the second expression is equal to 625 00:00 No. The first expression is equal to 625 and the second expression is equal to
Michael says that
[tex]5\cdot(\frac{1}{5^3})=5(5^3)[/tex]We are asked whether he is correct or not?
Let us simplify the equa
Find the equation that results from completing the square into the following equation X squared -14 X +40 equals zero
Given:
The equation is
[tex]x^2-14x+40=0[/tex]Required:
Find the equation that results from completing the square into the given equation.
Explanation:
The given equation is:
[tex]x^{2}-14x+40=0[/tex]Subtract 40 on both sides.
[tex]\begin{gathered} x^2-14x+40-40=0-40 \\ x^2-14x=-40 \end{gathered}[/tex]Add 49 on both sides.
[tex]\begin{gathered} x^2-14x+49=-40+49 \\ x^2-14x+49=9 \end{gathered}[/tex]Use the following formula:
[tex]a^2-2ab+b^2=(a-b)^2[/tex][tex](x-7)^2=9[/tex]Final answer:
The second option is the correct answer.
First, let's look at the measures of the length and width of the wall the boys arepainting. The problem says that the wall is 7 feet wide and 6 feet tall. Frank paints14 square feet of the wall, and Ryan paints the rest.How can you draw one straight line on this grid to divide it into two sections toshow the part of the wall Frank paints and the part of the wall Ryan paints?
Let's draw the 7ft by 6ft wall that is being painted by Frank and Ryan.
If Frank paints 14 square feet of the wall, we can say that Frank painted two rows of 7ft wide. See illustration below.
The darker horizontal line is the straight line that divides the wall into two sections.
So, the orange part is the section where Frank paints and the blank (white) part is the part of the wall that Ryan paints.
Graph the function and then determine the asymptote of the function.
We need to graph the following function:
[tex]y=\log _{1/2}(x)-4[/tex]We know that the domain of a logarithmic function can't be negative, then our domain is
[tex]x\in(0,\infty)[/tex]We need to analyze this function at its extremes to find the asymptotes.
Let's calculate the limit of this function at x = 0 and at infinity.
[tex]\begin{gathered} \lim _{x\to0}\log _{1/2}(x)-4=\infty \\ \lim _{x\to\infty}\log _{1/2}(x)-4=-\infty \end{gathered}[/tex]By definition, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. From our limits, we have an vertical asymptote at x =0 and no horizontal asymptotes.
This logarithmic function, comes from infinity at x = 0, an decays to minus infinity as x grows. We have the following graph:
A dairy needs 396 gallons of milk containing 6% butterfat. How many gallons each of milk containing 8% butterfat and milk containing 2% butterfat must be used to obtain the desired 396 gallons?
ANSWER:
264 gallons of milk containing 8% butterfat
132 gallons of milk containing 2% butterfat
STEP-BY-STEP EXPLANATION:
From the statement we can propose the following system of equations:
Let x be the milk that contains 8% butterfat and let y be the 2%.
[tex]\begin{gathered} x+y=396\rightarrow y=396-x\text{ (1)} \\ 0.08x+0.02y=0.06\cdot396\rightarrow0.08x+0.02y=23.76\text{ (2)} \end{gathered}[/tex]We substitute in equation (1) in equation (2) and substitute for x, just like this:
[tex]\begin{gathered} 0.08x+0.02\cdot(396-x)=23.76 \\ 0.08x+7.92-0.02x=23.76 \\ 0.06x=23.76-7.92 \\ x=\frac{15.84}{0.06} \\ x=264 \end{gathered}[/tex]Knowing the value of x, we can calculate the value of y, substituting in equation 1, like this:
[tex]\begin{gathered} y=396-264 \\ y=132 \end{gathered}[/tex]Therefore, 264 gallons of milk containing 8% butterfat and 132 gallons of milk containing 2% butterfat are needed.
Use add or sub formula to write as trig fun tho on
The formula to write this function is going to be:
[tex]tan(a-b)=\frac{tan(a)-tan(b)}{1+tan(a)*tan(b)}[/tex]Substituing:
[tex]\frac{tan(43)-tan(18)}{1+tan(43)tan(18)}=tan(43-18)[/tex]In this case a= 43 and b=18:
[tex]tan(43-18)=tan(25)=0.466\approx0.5[/tex]The answer is: tan(25).
The formula for calculating the distance, d, in miles that one can see to the horizon on a clear day is approximated by d=1.22radical x, where x is the elevation in feet of a person's eyes. a. approximately how far, to the nearest mile, can a person whose eyes are 600 feet above sea-level see? b. approximately how high, to the nearest foot, would a person's eyes need to be to see 100 miles?
The expression to calculate the distance a person can see is below, where x is the height in feet of a person's eyes above see-level:
[tex]d=\sqrt[1.22]{x}\lbrack mi\rbrack[/tex]a) A person who is 600 feet height will see:
[tex]d=\sqrt[1.22]{600}=189.30mi[/tex]b) In order to get the height a person needs to be so that he/she could see 100 miles long, we solve the equation for x:
[tex]\begin{gathered} d=x^{\frac{1}{1.22}} \\ d^{1.22}=(x^{\frac{1}{1.22}})^{1.22}=x \\ x=100^{1.22}=275.42ft \end{gathered}[/tex]Drag each label to the correct location on the table. Each label can be used more than once, but not all labels will be used.after constant its 6x squared -x-1
For polynomial 1: simplified form
[tex]6x^2-x-1[/tex]Name by Degree: Quadratic
Number of Terms: 3
Polynomial 2:
Simplified form is 3x+4
Name by Degree: Linear
Number of Terms: 2
Polynomial 3:
Simplified form: 2
Name by Degree: Number
Number of Terms: 1
[tex]3x+4[/tex]f(x) = -3x² + 6x + 1
find f(6)
Step-by-step explanation:
this is a college question ? and you don't know how to do this ?
come on, remember ! I don't know how you ever made it into college, if you don't understand this.
with f(6) we say x = 6, and now we simply have to put 6 into all the places of x and calculate.
f(6) = -3×6² + 6×6 + 1 = -3×36 + 36 + 1 = -2×36 + 1 =
= -72 + 1 = -71
Find the surface area and volume of the figure .The surface area is _ft2.(Round to the nearest tenth as needed .)
Question:
Find the surface area and volume of the figure.
Solution:
1) The surface area:
This shape is composed of a cylinder and hemisphere. Now, we know that the surface area of the sphere is:
[tex]SA\text{ sphere = 4}\pi\text{ }r^2[/tex]So that, the surface area of the hemisphere would be:
[tex]SA\text{ hemisphere = }2\pi r^2[/tex]On the other hand, the area of the circle is:
[tex]A\text{= }\pi r^2[/tex]thus, the surface area of the cylinder would be:
[tex]SA\text{ cylinder = }2\pi rh[/tex]replacing the data given in the problem in the formulas of the surface area of the hemisphere, area of the circle, and surface area of the cylinder, we get:
[tex]SA\text{ hemisphere = }2\pi(9)^2\text{ = 162}\pi[/tex]and
[tex]SA\text{ cylinder = }2\pi(9)(12)\text{ = }216\pi[/tex]and
[tex]A\text{= }\pi(9)^2=\text{ 81}\pi[/tex]then, we can conclude that the surface area of the given figure is:
[tex]SA\text{ = 162}\pi\text{ + 216}\pi+81\pi\text{ = 459}\pi\approx1441.9\text{ }\approx1442[/tex]that is:
[tex]SA\text{ }\approx1441.9\text{ }\approx1442[/tex]
2) The volume
The volume of a cylinder is given by the following formula:
[tex]V_C=\pi r^2h[/tex]and the volume of a hemisphere is :
[tex]V_H=\frac{1}{2}(\frac{4}{3}\pi r^3)\text{ = }\frac{2}{3}\pi r^3[/tex]thus, the volume of the figure would be:
[tex]V=V_C+V_H=\text{ }\pi r^2h\text{+}\frac{2}{3}\pi r^3[/tex]Then replacing the data given in the problem in the above formula we get:
[tex]V=\pi(9)^2(12)\text{+}\frac{2}{3}\pi(9)^3\text{ = 972}\pi\text{+486}\pi=\text{ 1458}\pi\approx4580.4\approx4580[/tex]that is;
[tex]V\approx4580.4\approx4580[/tex]Which graph represents 7x+2y<8? four different graphs to chose from.
we have the inequality
7x+2y < 8
the solution for this inequality is the shaded area below the dashed line 7x+2y=8
so
the slope of the dashed line is negative
the intercepts of the dashed line are
y-intercept is (0,4)
the x-intercept is (8/7,0)
therefore
the answer is option BHere are the numbers of times 13 people ate out last month 7, 3, 4, 3, 6, 4, 7, 6, 5, 7, 3, 6,5Find the modes of this data set.If there is more than one mode, write them separated by commas.If there is no mode, tap on "No mode."Explanation
In statistics, the mode is the value that occurs most frequently in a data set. This goes in the form of a column when we find two modes, that is, two data that have the same maximum absolute frequency.
First, we are going to count how many times each number is repeated in the data set.
[tex]\begin{gathered} 7\to3\text{ times} \\ 3\to3\text{ times} \\ 4\to2\text{ times} \\ 6\to3\text{ times} \\ 5\to2\text{ times} \end{gathered}[/tex]The highest frequency in the data set as we can see is 3. That is, the mode will be all the numbers that have that frequency in this case 7, 3 and 6