Question 3(Multiple Choice Worth 2 points)
(01.06 MC)
Simplify √√-72-
--6√√2
6√-2
6√√2i
061√2
Answers
Answer:
[tex]6i\sqrt{2}[/tex]
Step-by-step explanation:
Given expression:
[tex]\sqrt{-72}[/tex]
Rewrite -72 as the product of 6 · -1 · 2:
[tex]\implies \sqrt{36 \cdot -1 \cdot 2}[/tex]
Apply the radical rule [tex]\sqrt{ab}=\sqrt{a}\sqrt{b}:[/tex]
[tex]\implies \sqrt{36} \sqrt{-1} \sqrt{2}[/tex]
Carry out the square root of 36:
[tex]\implies 6\sqrt{-1}\sqrt{2}[/tex]
Apply the imaginary number rule [tex]\sqrt{-1}=i[/tex] :
[tex]\implies 6i\sqrt{2}[/tex]
Related Questions
A window washer drops a tool from their platform 155ft high. The polynomial -16t^2+155 tells us the height, in feet, of the tool t seconds after it was dropped. Find the height, in feet, after t= 1.5 seconds.
Answers
If you select one card at random from a standard deck of 52 cards, what is the probability of that card being a 5, 6 OR 7?
Answers
To solve this question we will use the following expression to compute the theoretical probability:
[tex]\frac{\text{favorable cases}}{total\text{ cases}}.[/tex]1) We know that there are 4 fives, 4 sixes, and 4 sevens in a standard deck of 52 cards, then, the probability of selecting a 5, 6, or 7 is:
[tex]\frac{4+4+4}{52}\text{.}[/tex]2) Simplifying the above expression we get:
[tex]\frac{12}{52}=\frac{3}{13}\text{.}[/tex]Answer:
[tex]\frac{3}{13}\text{.}[/tex]A pound of rice crackers cost 42.88 Jacob purchased a 1/4 pound how much did he pay for the crackers?
Answers
Answer:
10.72
Step-by-step explanation:
The price per pound is 42.88
We are getting 1/4 pound.
Multiply 42.88 by 1/4
42.88 * 1/4 =10.72
Answer:
So you know that a pound of rice crackers cost $42.88. You also know that Matthew bought 1/4 or 25% or 0.25 of a pound. This means that by 42.88 divided 4 will equal the answer.
42.88 ÷ 4 = 10.72
Therefore, Matthew paid or $10.72 for 1/4 pound of rice crackers.
What is the first step for finding the quotient of 3x^3 z^5/5y * x^2 z^6/20y^3
Answers
The initial expression is:
[tex]\frac{3x^3z^5}{5y}\text{ / }\frac{x^2z^6}{20y^3}[/tex]So the first step is to multiply the numerator of the second fraction with the denominator of the first franction and the denominator of the second fraction by the numerator of the first fraction so:
[tex]\frac{3x^3z^6}{5y}(\frac{20y^3}{x^2z^6})[/tex]So is option C)
Help asp show your work you’ll get brainliest
Answers
The information given in the table on the Value of a Car and the Age of the Car, gives;
First Part;
The dependent variable is; The Value of Car
The independent variable is; The Age of Car
Second part;
The situation is a function given that each Age of Car maps to only one Value of Car.
What is a dependent and a independent variable?A dependent variable is an output variable which is being observed, while an independent variable is the input variable which is known or controlled by the researcher.
First part;
The given information in the table is with regards to how the car's value decreases with time, therefore;
The dependent variable, which is the output variable, or the variable whose value is required is the current Value of the Car (Dollars)The independent variable, which is the input variable, or the variable that determines the value of the output or dependent variable, is the Age of Car (Years)Second part;
A function is a relationship in which each input value has exactly one output.
Given that the Values of the cars are all different, and no two car of a particular age has two values, therefore;
The situation is a functionGiven that the first difference varies depending on the age of the car, the function can be taken as a piecewise function
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the length of a rectangle is 13 centimeters less then four times it’s width it’s area is 35 centimeters find the dimensions of the rectangle
Answers
Solution:
The area of a recatngle is expressed as
[tex]\begin{gathered} \text{Area of rectangle = L}\times W \\ \text{where} \\ L\Rightarrow\text{length of the rectangle} \\ W\Rightarrow\text{ width of the rectangle } \end{gathered}[/tex]Given that the length of the rectangle is 13 centimeters less than four times its width, this implies that
[tex]L=4W-13\text{ ---- equation 1}[/tex]Tha area of the rectangle is 35 square centimeters. This implies that
[tex]36=L\times W\text{ --- equation 2}[/tex]Substitute equation 1 into equation 2. Thus,
[tex]\begin{gathered} 36=L\times W \\ \text{where} \\ L=4W-13 \\ \text{thus,} \\ 36=W(4W-13) \\ open\text{ parentheses} \\ 36=4W^2-13W \\ \Rightarrow4W^2-13W-36=0\text{ ---- equation 3} \\ \end{gathered}[/tex]Solve equation 3 by using the quadratic formula expressed as
[tex]\begin{gathered} W=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}_{} \\ \text{where} \\ a=4 \\ b=-13 \\ c=-36 \end{gathered}[/tex]thus, we have
[tex]\begin{gathered} W=\frac{-(-13)\pm\sqrt[]{(-13)^2-(4\times4\times-36)}}{2\times4}_{} \\ =\frac{13\pm\sqrt[]{169+576}}{8} \\ =\frac{13\pm\sqrt[]{745}}{8} \\ =\frac{13}{8}\pm\frac{\sqrt[]{745}}{8} \\ =1.625\pm3.411836016 \\ \text{thus,} \\ W=5.036836016\text{ or W=}-1.786836016 \end{gathered}[/tex]but the width cannot be negative. thus, the width of the recangle is
[tex]W=5.036836016[/tex]From equation 1,
[tex]\begin{gathered} L=4W-13 \\ \end{gathered}[/tex]substitute the obtained value of W into equation 1.
Thus, we have
[tex]\begin{gathered} L=4W-13 \\ =4(5.036836016)-13 \\ =20.14734-13 \\ \Rightarrow L=7.14734 \end{gathered}[/tex]Hence:
The width is
[tex]5.036836016cm[/tex]The length is
[tex]7.14734cm[/tex]Find the absolute maximum and minimum values of the following function on the given interval. f(x)=3x−6cos(x), [−π,π]
Answers
Answer:
Absolute minimum: x = -π / 6
Absolute maximum: x = π
Explanation:
The candidates for the absolute maximum and minimum are the endpoints and the critical points of the function.
First, we evaluate the function at the endpoints.
At x = -π, we have
[tex]f(-\pi)=3(-\pi)-6\cos (-\pi)[/tex][tex]\Rightarrow\boxed{f(-\pi)\approx-3.425}[/tex]At x = π, we have
[tex]f(\pi)=3(\pi)-6\cos (\pi)[/tex][tex]\Rightarrow\boxed{f(\pi)\approx15.425.}[/tex]Next, we find the critical points and evaluate the function at them.
The critical points = are points where the first derivative of the function are zero.
Taking the first derivative of the function gives
[tex]\frac{df(x)}{dx}=\frac{d}{dx}\lbrack3x-6\cos (x)\rbrack[/tex][tex]\Rightarrow\frac{df(x)}{dx}=3+6\sin (x)[/tex]Now the critical points are where df(x)/dx =0; therefore, we solve
[tex]3+6\sin (x)=0[/tex]solving for x gives
[tex]\begin{gathered} \sin (x)=-\frac{1}{2} \\ x=\sin ^{-1}(-\frac{1}{2}) \end{gathered}[/tex][tex]x=-\frac{\pi}{6},\; x=-\frac{5\pi}{6}[/tex]
on the interval [−π,π].
Now, we evaluate the function at the critical points.
At x = -π/ 6, we have
[tex]f(-\frac{\pi}{6})=3(-\frac{\pi}{6})-6\cos (-\frac{\pi}{6})[/tex][tex]\boxed{f(-\frac{\pi}{6})\approx-6.77.}[/tex]At x = -5π/6, we have
[tex]f(\frac{-5\pi}{6})=3(-\frac{5\pi}{6})-6\cos (-\frac{5\pi}{6})[/tex][tex]\Rightarrow\boxed{f(-\frac{5\pi}{6})\approx-2.66}[/tex]Hence, our candidates for absolute extrema are
[tex]\begin{gathered} f(-\pi)\approx-3.425 \\ f(\pi)\approx15.425 \\ f(-\frac{\pi}{6})\approx-6.77 \\ f(-\frac{5\pi}{6})\approx-2.66 \end{gathered}[/tex]Looking at the above we see that the absolute maximum occurs at x = π and the absolute minimum x = -π/6.
Hence,
Absolute maximum: x = π
Absolute minimum: x = -π / 6
Use the remainder theorem to find P(-2) for P(x) = x³ + 3x² +9,Specifically, give the quotient and the remainder for the associated division and the value of P(-2).QuotientRemainder =P(-2)=
Answers
Answer:
Quotient:
[tex]x^2+x-2[/tex]Remainder:
[tex]13[/tex]P(-2):
[tex]13[/tex]Step-by-step explanation:
Remember that the remainder theorem states that the remainder when a polynomial p(x) is divided by (x - a) is p(a).
To calculate the quotient, we'll do the synthetic division as following:
Step one:
Write down the first coefficient without changes
Step two:
Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).
Add the obtained result to the next coefficient of the dividend, and write down the sum.
Step 3:
Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).
Add the obtained result to the next coefficient of the dividend, and write down the sum.
Step 4:
Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).
Add the obtained result to the next coefficient of the dividend, and write down the sum.
Now, we will have completed the division and have obtained the following resulting coefficients:
[tex]1,1,-2,13[/tex]Thus, we can conlcude that the quotient is:
[tex]x^2+x-2[/tex]And the remainder is 13, which is indeed P(-2)
help meeeeeeeeee pleaseee !!!!!
Answers
The composition of the function (g o f)(5) is evaluated as: (g o f)(5) = g(f(5)) = 6.
How to Determine the Composition of a Function?To find the composition of a function, we have to first evaluate the inner function for the given value of x that is given as its input. After that, the output of the inner function would then be used as the input for the outer function, which would now be evaluated for the composition of the function.
Given the functions:
f(x) = x² - 6x + 2
g(x) = -2x
We need to find the composition of the function, (g o f)(5), where the inner function is f(x), and the outer function is g(x).
Therefore:
(g o f)(5) = g(f(5))
Find f(5):
f(5) = (5)² - 6(5) + 2
f(5) = -3
Substitute x = -3 into g(x) = -2x:
(g o f)(5) = -2(-3)
(g o f)(5) = 6
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you bought a car for $5000. each year it depreciates by 8.5%. Which equation can be used to find the value, v, of the car, x years after it was purchased?
Answers
We have the following:
In this case, we have the following formula:
[tex]v=C\cdot(1-r)^x[/tex]Where C is the original value of the car, r is the depreciation rate and x is the time in years
Can someone help with this question?✨
Answers
The equation of the line that is perpendicular with y = 4 · x - 3 and passes through the point (- 12, 7) is y = - (1 / 4) · x + 4.
How to derive the equation of a line
In this problem we find the case of a line that is perpendicular to another line and that passes through a given point. The equation of the line in slope-intercept form is described below:
y = m · x + b
Where:
m - Slopeb - Interceptx - Independent variable.y - Dependent variable.In accordance with analytical geometry, the relationship between the two slopes of the lines are:
m · m' = - 1
Where:
m - Slope of the first line.m' - Slope of the perpendicular line.If we know that m = 4 and (x, y) = (- 12, 7), then the equation of the perpendicular line is:
m' = - 1 / 4
b = 7 - (- 1 / 4) · (- 12)
b = 7 + (1 / 4) · (- 12)
b = 7 - 3
b = 4
And the equation of the line is y = - (1 / 4) · x + 4.
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Find the formula for an exponential function that passes through the 2 points given
Answers
The form of the exponential function is
[tex]f(x)=a(b)^x[/tex]a is the initial value (value f(x) at x = 0)
b is the growth/decay factor
Since the function has points (0, 6) and (3, 48), then
Substitute x by 0 and f(x) by 6 to find the value of a
[tex]\begin{gathered} x=0,f(x)=6 \\ 6=a(b)^0 \\ (b)^0=1 \\ 6=a(1) \\ 6=a \end{gathered}[/tex]Substitute the value of a in the equation above
[tex]f(x)=6(b)^x[/tex]Now, we will use the 2nd point
Substitute x by 3 and f(x) by 48
[tex]\begin{gathered} x=3,f(x)=48 \\ 48=6(b)^3 \end{gathered}[/tex]Divide both sides by 6
[tex]\begin{gathered} \frac{48}{6}=\frac{6(b)^3}{6} \\ 8=b^3 \end{gathered}[/tex]Since 8 = 2 x 2 x 2, then
[tex]8=2^3[/tex]Change 8 to 2^3
[tex]2^3=b^3[/tex]Since the powers are equal then the bases must be equal
[tex]2=b[/tex]Substitute the value of b in the function
[tex]f(x)=6(2)^x[/tex]The answer is:
The formula of the exponential function is
[tex]f(x)=6(2)^x[/tex]a janitor had 2/3 of a cleaning solution. he used 1/4 of the solution in an day. how much of the bottle did he use?
Answers
Answer:
5/12 of the cleaning solution.
Step-by-step explanation:
2/3 – 1/4
------------------------------------------
2 × 4
= 8/12
3 × 4
------------------------------------------
1 x 3
= 3/12
4 x 3
------------------------------------------
8 – 3
12
= 5/12
------------------------------------------
Hopefully this makes sense!
A circle has a circumference of 10 inches. Find its approximate radius, diameter and area
Answers
Answer:
Radius = 1.59 in
Diameter = 3.18 in
Area = 7.94 in²
Explanation:
The circumference of a circle can be calculated as:
[tex]C=2\pi r[/tex]Where r is the radius of the circle and π is approximately 3.14. So, replacing C by 10 in and solving for r, we get:
[tex]\begin{gathered} 10\text{ in = 2}\pi r \\ \frac{10\text{ in}}{2\pi}=\frac{2\pi r}{2\pi} \\ 1.59\text{ in = r} \end{gathered}[/tex]Then, the radius is 1.59 in.
Now, the diameter is twice the radius, so the diameter is equal to:
Diameter = 2 x r = 2 x 1.59 in = 3.18 in
On the other hand, the area can be calculated as:
[tex]A=\pi\cdot r^2[/tex]So, replacing r = 1.59 in, we get:
[tex]\begin{gathered} A=3.14\times(1.59)^2 \\ A=3.14\times2.53 \\ A=7.94in^2 \end{gathered}[/tex]Therefore, the answer are:
Radius = 1.59 in
Diameter = 3.18 in
Area = 7.94 in²
f(x)A6X-868Which of the given functions could this graph represent?OA. f(t) = (x - 1)(x - 2)(x + 1)(x + 2)O B. f(x) = x(x - 1)(1 + 1)Oc. /(x) = x(x - 1)(x - 2)(x + 1)(x + 2)OD. (r) = x(x - 1)(x - 2)
Answers
The Solution:
Given the graph below:
We are required to determine the function that best describes the above graph.
Step1:
Identify the roots of the function from the given graph.
[tex]\begin{gathered} x=-2 \\ x=-1 \\ x=1 \\ x=2 \end{gathered}[/tex]This means that:
[tex]\begin{gathered} x+2=0 \\ x+1=0 \\ x-1=0 \\ x-2=0 \end{gathered}[/tex]So, the required function becomes:
[tex]f(x)=(x-1)(x-2)(x+1)(x+2)[/tex]Therefore, the correct answer is [option A]
Find the missing rational expression.382x + 6(x-3)(x + 1)X-332x + 6(x-3)(x + 1)(Simplify your answer.)X-3
Answers
Which number is greater in each set?
Answers
We have three set of numbers and we must choose the greater value in each set
1.
[tex]\frac{1}{3}or\frac{1}{4}or\frac{1}{5}[/tex]When the numerator is 1, the greater fraction is the one that has the small denominator.
So, in this case the greater number is
[tex]\frac{1}{3}[/tex]2.
[tex]\frac{1}{4}or\frac{4}{3}or\frac{5}{6}[/tex]In this case we can rewrite the fractions as fractions with the same denominator
[tex]\frac{1}{4}=\frac{3}{12}[/tex][tex]\frac{4}{3}=\frac{16}{12}[/tex][tex]\frac{5}{6}=\frac{10}{12}[/tex]Then, the greater number is the one that has the greater numarator
So, it is
[tex]\frac{16}{12}=\frac{4}{3}[/tex]in this case the greater number is
[tex]\frac{4}{3}[/tex]3.
[tex]\frac{16}{5}or3\frac{2}{5}or3.25[/tex]In this case we can rewrite the numbers as decimal numbers
[tex]\frac{16}{5}=3.2[/tex][tex]3\frac{2}{5}=3.4[/tex][tex]3.25=3.25[/tex]In this case the greater number is
[tex]3\frac{2}{5}[/tex]Need help !! Geometry unit 3 parallel and perpendicular lines
Answers
ANSWER;
Converse; Exterior alternate angles are equal
[tex]x\text{ = 3}[/tex]EXPLANATION;
Here, we want to get the value of x given that the lines l and m are parallel
From the diagram given, we can see that;
[tex]15x\text{ +29 = 26x-4}[/tex]The reason for this is that they are a pair of exterior alternate angles
Mathematically, exterior alternate angles are equal
From here, we can proceed to solve for the value of x;
[tex]\begin{gathered} 26x-15x\text{ = 29+4} \\ 11x=33 \\ x\text{ = }\frac{33}{11} \\ x\text{ = 3} \end{gathered}[/tex]Which of the following statements must be true based on the diagram below!(Diagram is not to scale)O JL is a segment bisector.JL is a perpendicular bisector.OJT is an angle bisectora Lis the vertex of a right angle,Jis the midpoint of a segment in the diagramNone of the above.
Answers
From the diagram, we notice that the line JL bisects the angle J into two equal angles. Hence, we can conclude that the correct statement is this:
JL is an angle bisector
An angle bisector are
ur answer as a polynomial in standard form.=f(x) = 5x + 1g(x) = x2 – 3x + 12=Find: (fog)(x)
Answers
(fog)(x) = 5x² - 15x + 61
Explanation:The given functions are:
f(x) = 5x + 1
g(x) = x² - 3x + 12
(fog)(x) = f(g(x))
This means that we are substituting g(x) into f(x)
(fog)(x) = 5(x² - 3x + 12) + 1
(fog)(x) = 5x² - 15x + 60 + 1
This can be further simplified as:
(fog)(x) = 5x² - 15x + 61
Angle RQT is a straight angle. What are m angle RQS and m angle TQS? Show your work.
Answers
11x + 5 + 8x + 4 = 180
Simplifying like terms
11x + 8x = 180 - 5 - 4
19x = 171
x = 171/19
x = 9
RQS = 11(9) + 5
= 99 + 5
= 104°
TQS = 8(9) + 4
= 72 + 4
= 76°
If each machine produces nails at the same rate, how many nails can 1 machine produce in 1 hour
Answers
Divide the number of nails by the number of minutes:
16 1/5 ÷ 15 = 1 2/25 per minute
48 3/5 ÷ 45 = 1 2/25 per min
59 2/5 ÷ 55 = 1 2/25 per min
We have the number of nails produced per minute, to calculate the number of nails in an hour multiply it by 60, because 60 minutes= 1 hour:
1 2/25 x 60 = 64 4/5
helppppppppppppppppppp
Answers
Don’t get part b of the question. Very confusing any chance you may help me with this please.
Answers
To solve this problem, first, we will solve the given equation for y:
[tex]\begin{gathered} x=3\tan 2y, \\ \tan 2y=\frac{x}{3}, \\ 2y=\arctan (\frac{x}{3}), \\ y=\frac{\arctan(\frac{x}{3})}{2}=\frac{1}{2}\arctan (\frac{x}{3})\text{.} \end{gathered}[/tex]Once we have the above equation, now we compute the derivative. To compute the derivative we will use the following properties of derivatives:
[tex]\begin{gathered} \frac{d}{dx}\arctan (x)=\frac{1}{x^2+1}, \\ \frac{dkf(x)}{dx}=k\frac{df(x)}{dx}. \end{gathered}[/tex]Where k is a constant.
First, we use the second property above, and get that:
[tex]\frac{d\frac{\arctan(\frac{x}{3})}{2}}{dx}=\frac{d\arctan (\frac{x}{3})\times\frac{1}{2}}{dx}=\frac{1}{2}\frac{d\arctan (\frac{x}{3})}{dx}\text{.}[/tex]Now, from the chain rule, we get:
[tex]\frac{dy}{dx}=\frac{1}{2}\frac{d\text{ arctan(}\frac{x}{3})}{dx}=\frac{1}{2}\frac{d\arctan (\frac{x}{3})}{dx}|_{\frac{x}{3}}\frac{d\frac{x}{3}}{dx}\text{.}[/tex]Finally, computing the above derivatives (using the rule for the arctan), we get:
[tex]\frac{dy}{dx}=\frac{1}{2}\frac{\frac{1}{3}}{\frac{x^2}{9}+1}=\frac{1}{6}(\frac{1}{\frac{x^2}{9}+1})=\frac{3}{2(x^2+9)}.[/tex]Answer:
[tex]\frac{3}{2(x^2+9)}.[/tex]For p(2) = 7 + 10x - 12x^2 - 10x^3 + 2x^4 + 3x^5, use synthetic substitution to evaluate
Answers
Answer:
p(-3) = -428
Explanations:Given the polynomial function expressed as:
[tex]p(x)=7+10x-12x^2-10x^3+2x^4+3x^5[/tex]Determine the value of p(-3)
[tex]\begin{gathered} p(-3)=7+10(-3)-12(-3)_^2-10(-3)^3+2(-3)^4+3(-3)^5 \\ p(-3)=7-30-12(9)-10(-27)+2(81)+3(-243) \\ p(-3)=-23-108+270+162-729 \\ p(-3)=-428 \end{gathered}[/tex]Hence the value of p(-3) is -428
Need help with this question
Answers
Given: a quadratic function with vertex (2,3) opening upward .
Find: the given statement is true or false.
Explanation: if parabola has a vertex at (2,3) and opens upward, it has one real solution., (2,3) will be a lowest point. The vertex will be at lowest point, it will be minimum.
that means graph has no one real solution. hence it will never going to intersect. so this statement is false.
Final answer: the given statement is FALSE.
2x - 6(x-3) ≥ 5
solve for x.
Answers
Answer:
It’s siu
Step-by-step explanation:
Answer:x≤4.6
Step-by-step explanation: 2x-6(x-3)≥5. 1).combine the like terms. 2x+x=3x & -6+-3=-9. 2). isolate the "x". 3x-9≥5. 3x≥14. 3). divide both sides by your coefficient. 3x≥14/ 3
x≥4.6
4) flip your sign. x≤4.6
The graph of f(x) is shown in black.Write an equation in terms of f(x) to match the redgraph.For example, try something like this:f(x)+3, f (x - 2), or 4f(x).
Answers
Notice that the red function is similar to the black function, which means the transformation applied was a translation.
The transformation is 5 units to the right, exactly.
Therefore, the function that represents the red figure is
[tex]f(x-5)[/tex]1. Sketch the graph of y = x that is stretched vertically by a factor of 3. (Hint: Write the equation first, then graph) Sketch both y = x and the transformed graph.
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ANSWER and EXPLANATION
We want to stretch the graph of:
y = x
A vertical stretch of a linear function is represented as:
y' = c * y
where c is the factor
The factor from the question is 3.
So, the new equation is:
y' = 3 * x
y' = 3x
Let us plot the functions:
12 = - 2/5 yI got -30 I want to see if I did the correct steps
Answers
Solution
[tex]12=-\frac{2}{5}y[/tex]Step 1: Simplify the expression
[tex]\begin{gathered} 12=-\frac{2}{5}y \\ \text{cross multiply} \\ 12(5)=-2y \\ 60=-2y \end{gathered}[/tex]Step 2: Divide the both side by -2
[tex]\begin{gathered} 60=-2y \\ \frac{60}{-2}=-\frac{2y}{-2} \\ y=-30 \end{gathered}[/tex]Therefore the correct value of y = - 30