Final answer:
The square root of -9 is not -3 but 3i, indicating the need for complex numbers to describe the square root of negative numbers, with 'i' denoting the imaginary unit.
Explanation:
The value of the square root of -9 is not -3 because the square root operation defined for real numbers does not apply to negative numbers in the same way. Instead, when dealing with the square root of a negative number, we enter the realm of complex numbers.
A complex number is a combination of a real part and an imaginary part. The square root of -1 is defined as i, the imaginary unit. Therefore, the square root of -9 is correctly expressed as 3i, not -3. This is a critical distinction because the result provides a basis for complex arithmetic, expanding the scope of mathematical operations into the complex plane.
Complex numbers and imaginary numbers are fundamental in various fields of science and engineering, particularly in situations involving quadratic equations with no real roots or in the analysis of electrical circuits. Recognizing the utility of the imaginary unit, i, allows for solving equations that otherwise would not have solutions within the set of real numbers.
find the equation of a line with the given points.(put your answers in the form y=mx+b)(6,2)(5,5)
The slope-intercept form: y= mx + b
m - slope
b - y-intercept
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
We have the points (6, 2) and (5, 5). Substitute:
[tex]m=\dfrac{5-2}{5-6}=\dfrac{3}{-1}=-3[/tex]
Therefore we have y = -3x + b.
Put the coordinates of the point (5, 5) to the equation:
5 = -3(5) + b
5 = -15 + b add 15 to both sides
20 = b
Answer: y = -3x + 20Answer:
y = -3x + 20
Step-by-step explanation:
(a) Slope
The point-slope formula for a straight line is
y₂ - y₁ = m(x₂ - x₁) Insert the points
2 - 5 = m(6 - 5)
-3 = m×1
m = -3
=====
(b) y-intercept
y₂ - y₁ = m(x₂ - x₁)
y₂ - 5 = -3(x₂ - 5) Remove parentheses
y₂ - 5 = -3x₂ - 15 Add 5 to each side
y = -3x + 20
Karla, Ruby, Anna, and Megan each had a balance of $0 in her bank account on the first day of April. Here are their account balances on the second day:
Karla: $43
Ruby: -$25
Anna: -$48
Megan: $42
Whose account had the greatest change between the first and second days?
A.
Karla's
B.
Ruby's
C.
Anna's
D.
Megan's
PLZZ Healp
Answer:
Anna
Step-by-step explanation:
Compare the answers, and when you do that then you can see that anna had more money come in then the others.
A 3 mi cab ride cost $3.00 . A 6 mi can ride cost $4.80 find a linear equation that modes cost c as a function of distance d.
Answer:
c = 0.6d + 1.2
Step-by-step explanation:
Use distance as the independent variable and cost as the dependent variable. Each point (x, y) is (distance, cost).
The 3 mile ride for $3.00 gives you point (3, 3).
The 6 mile ride that cost $4.80 gives you point (6, 4.8).
Now you need to find the equation of the line that passes through points
(3, 3) and (6, 4.8).
y = mx + b
m = slope = (y2 - y1)/(x2 - x1) = (4.8 - 3)/(6 - 3) = 1.8/3 = 0.6
Use point (3, 3).
y = 0.6x + b
3 = 0.6(3) + b
3 = 1.8 + b
b = 1.2
The equation is
y = 0.6x + 1.2
Use c for cost and d for distance to get
c = 0.6d + 1.2
Need answers for 14-17
use the function below to find f(–2). f(x) = 3^x.
When an exponent is negative, you move it to the other side of the fraction to make the exponent positive.
For example:
[tex]x^{-2}[/tex] or [tex]\frac{x^{-2}}{1} =\frac{1}{x^2}[/tex]
[tex]\frac{1}{y^{-3}} =\frac{y^3}{1}[/tex] or y³
f(-2) This means that x is -2, so you can plug in -2 for "x" in the equation
[tex]f(x)=3^x[/tex]
[tex]f(-2)=3^{-2}[/tex]
[tex]f(-2)=\frac{1}{3^2}[/tex]
[tex]f(-2)=\frac{1}{9}[/tex]
Your answer is D
The value of the function f(-2) will be 1/9
What is a function?A mathematical relationship from a set of inputs to a set of outputs is called a function.
How to find the value of f(-2) ?The given function is f(x) = [tex]3^{x}[/tex]
To find f(-2) , we will have to put -2 in place of x∴ f(-2) will be = [tex]3^{-2}[/tex]
Now, we know that [tex]a^{-b}[/tex] can be written as [tex]\frac{1}{a^{b} }[/tex]So, [tex]3^{-2}[/tex] will be equal to 1/9
∴ The value of f(-2) will be 1/9
Option D is correct.
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x-1/8=5/24
A . 1/12
B. 5/24
C. 1/3
D. 1 2/3
Answer:
The answer c
Hope I helped :)
Step-by-step explanation:
Solve for x by simplifying both sides of the equation then isolating the variable.
Exact Form:
x=1/3
Decimal Form:
x=0.333333333
On an architects blueprint, 1 inch corresponds to 4 feet. Find the length of a wall represented by a line that is 3 7/8 inches long on the blueprint
To find the length of the wall represented by a line that is 3 7/8 inches long on the blueprint, we can use the scale conversion. The length of the wall is 15 1/2 feet.
Explanation:To find the length of the wall represented by a line that is 3 7/8 inches long on the blueprint, we need to use the scale conversion.
The given scale is 1 inch corresponds to 4 feet.
So, we can set up a proportion:
1 inch − 4 feet
3 7/8 inches − x feet
To solve for x, we can cross-multiply:
1 × x = 4 × 3 7/8
Multiply 4 by the whole number 3 and then add the product to the result of 4 multiplied by the fractional part 7/8.
x = 12 + 28/8
x = 12 + 3 1/2
x = 15 1/2 feet
Therefore, the length of the wall represented by a line that is 3 7/8 inches long on the blueprint is 15 1/2 feet.
The box plots below show the ages of college students in different math courses
Answer:
The statement that most accurately represents the data given is the 'The mean and median age are more likely to be the same for the students in Math 1.'
Step-by-step explanation:
The box and whisker plots given for the two sets of data show a few different things. Box and whisker plots first show us the median in a given set of data, which is the line in the middle of the box. For both Math 1 and Math 2, this line occurs at 19. The whiskers that extend from the box on either side represent the range, or lowest and highest numbers, in the data. For Math 1, the range is 17-21 and for Math 2, the range is 17-23. Since the range is more evenly distributed in Math 1, we can conclude that the mean, or average of the data, is most likely the same as the median of the data, which is 19.
Answer:
its d
Step-by-step explanation:
its d
Roger Ratkin, the owner of Roger’s Subs, has three employees who earn $500, $550, and $700, respectively. How much does Roger owe for the first 12 weeks for SUTA and FUTA? Assume a SUTA rate of 5.3% and a FUTA rate of .8%.
To calculate SUTA and FUTA for the first 12 weeks, calculate the taxable wages for each employee. Multiply the taxable wages by the respective tax rates to find the amount owed. Roger owes $371 for SUTA and $56 for FUTA for the first 12 weeks.
Explanation:To calculate the amount that Roger owes for SUTA (State Unemployment Tax Act) and FUTA (Federal Unemployment Tax Act) for the first 12 weeks, we need to calculate the taxable wages for each employee. The taxable wages for SUTA is the first $7,000 and for FUTA is the first $7,000 as well.
Since the three employees earn $500, $550, and $700 respectively, their total wages for the first 12 weeks would be $17,100 (12 weeks x ($500 + $550 + $700)). However, since SUTA and FUTA only consider the first $7,000 of wages, we can calculate the amount for each tax.
For SUTA, the total taxable wages would be $7,000. Using the SUTA rate of 5.3%, we can calculate the amount owed: $7,000 x 0.053 = $371.
For FUTA, the total taxable wages would also be $7,000. Using the FUTA rate of 0.8%, we can calculate the amount owed: $7,000 x 0.008 = $56.
Therefore, Roger owes $371 for SUTA and $56 for FUTA for the first 12 weeks for his three employees.
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Teresa makes brownies in the oven for 55 minutes. She checks on the brownies after 36 minutes. How much longer do the brownies need to bake?
Answer:
19 minutes.
Step-by-step explanation:
55-36=19.
For this case we have to indicate how much time the brownies are needed to bake, knowing that 55 minutes are given and Teresa reviewed them after 36 minutes.
For this we subtract:
[tex]55 \ minutes-36 \ minutes = 19 \ minutes[/tex]
So brownies have 19 minutes to bake
Answer:
19 minutes
Ari stocks shelves at a grocery store.He puts 35 cans of juice in each display case.The case has 4 shelves with an equal number of cans,and one shelf with only 3 cans.How many cans are on each of the equal shelves?
Answer:
There are 8 cans on each of the equal shelves
Step-by-step explanation:
Total number of cans put in display = 35
Shelves with equal no of cans = 4 Shelves
And one shelf displays only 3 cans
Therefore, 4 shelves with equal no of cans + 1 shelf with 3 cans = 35 cans
Let us assume equal no. of cans be x
We get, 4x + 3 = 35
4x = 35 - 3
4x = 32
x = 32 ÷ 4
x = 8
Thus, 4 shelves with equal no of cans contains 8 cans each.
i dont really know how to do this
Answer: x^3y^4 is the lowest common denominator.
Step-by-step explanation: To get both expressions to have a common denominator, multiply both of them to get them equal. For the first expression, multiply the numerator and denominator by y^3: 3y^3/x^3y^4
The second expression should be multiplied by the numerator and denominator by x^2: 7x^2/x^3y^4
(Add exponents when multiplying them) Now both expressions have common denominators.
Answer:
x^3*y^4
Step-by-step explanation:
The two denominators shown are x^3*y and x*y^4. The LCD must involve the largest power of x, which is x^3, and the largest power of y, which is y^4. Thus, the LCD is x^3*y^4.
Consider functions f and g below.
f(x) = -9x^2 - 7x + 12
g(x) = 3x^2 - 4x - 15
Find f(x) - g(x).
A. -12x^2 - 11x - 3
B. 6x^2 - 11x + 27
C. -6x^2 + 3x - 3
D. -12x^2 - 3x + 27
Answer:
D
Step-by-step explanation:
f(x) - g(x) = -9x^2 - 7x + 12 - (3x^2 - 4x - 15) Remove the brackets
f(x) - g(x) = -9x^2 - 7x + 12 - 3x^2 + 4x + 15
f(x) - g(x) = -9x^2 - 3x^2 - 7x + 4x + 12 + 15
f(x) - g(x) = -12x^2- 3x + 27
Answer D
Answer:
Option D. -12x² - 3x + 27
Step-by-step explanation:
The given functions are f(x) = -9x² - 7x + 12 and g(x) = 3x² - 4x - 15
We have to find the value of f(x) - g(x)
f(x) - g(x) = (-9x² - 7x + 12) - (3x² - 4x - 15)
= -9x² - 7x + 12 - 3x² + 4x + 15
= (-9x² - 3x²) - (7x - 4x) + 12 + 15
= -12x² - 3x + 27
This value of f(x) - g(x) matches with option D.
Therefore, Option D. is the correct option.
What is the slope and y-intercept of the line represented by the equation below? 13x+8y=12
Answer:
see explanation
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y-intercept )
rearrange 13x + 8y = 12 into this form
subtract 13x from both sides
8y = - 13x + 12 ( divide all terms by 8 )
y = - [tex]\frac{13}{8}[/tex] + [tex]\frac{3}{2}[/tex] ← in slope-intercept form
with slope m = - [tex]\frac{13}{8}[/tex] and y-intercept c = [tex]\frac{3}{2}[/tex]
I'm doing test corrections rn but I need to show the work for this answer. The problem is that the this doesn't show the work and I need it for full credit, can someone show me the work please?
The two angles are corresponding angles, so they are congruent (due to the horizontal lines being parallel)
x+15 = 102
x+15-15 = 102-15 .... subtract 15 from both sides
x = 87
Which models can be used to solve the problem
Which of the following expressions is equal to 2x^(2)+8?
(2x-4i)(x-2i)
(2x-4i)(x-2i)
(2x+4i)(x+2i)
(2x_2i)(x+6i)
Answer:
The expression equal to 2x^(2)+8 is:
First option (2x-4i)(x+2i)
Step-by-step explanation:
If we multiply
(2x-4i)(x+2i)=(2x)(x)+(2x)(2i)-(4i)(x)-4i(2i)
(2x-4i)(x+2i)=2x^(1+1)+4xi-4xi-8i^(1+1)
Simplifying:
(2x-4i)(x+2i)=2x^2-8i^2
and i^2=-1, then:
(2x-4i)(x+2i)=2x^2-8(-1)
Multiplying:
(2x-4i)(x+2i)=2x^2+8
Answer: (2x-4i)(x+2i)
Step-by-step explanation:
What is 3.27 × 10−4 in standard form? A: 0.0000327
B: 0.000327
C: 0.0327
D: 3,270
Answer:
b
Step-by-step explanation:
3.27 Times 10⁻⁴ in standard form is 0.000327. This is found by moving the decimal point in 3.27 four places to the left, resulting in the number 0.000327.
The number 3.27 TImes 10⁻⁴ into standard form.
To do this, we move the decimal point four places to the left, because the exponent on the 10 is negative four.
So, starting with the number 3.27, we move the decimal point to the left four times, which introduces three zeros after the decimal point and before the 327.
The correct answer is B: 0.000327.
This means that 3.27 times 10⁻⁴ is equivalent to 0.000327 when written in standard form.
The sum of two numbers is 13. Two times the first number minus three times the second number is 1. If you let x stand for the first number and y for the second number what are the two numbers
So for this, we will be setting up a system of equations with the information we have:
[tex]x+y=13\ \textsf{"The sum of two numbers is 13."}\\2x-3y=1\ \textsf{"Two times the first number minus three times the second number is 1."}[/tex]
Now we have our system of equations set up. Next, I will be using the substitution method to solve this system. So firstly, subtract y on both sides of the first equation:
[tex]x=13-y\\2x-3y=1[/tex]
Now, substitute x for (13 - y) in the second equation and solve for y as such:
[tex]2(13-y)-3y=1\\26-2y-3y=1\\26-5y=1\\-5y=-25\\y=5[/tex]
Now that we have the value of y, substitute it into either equation to solve for x:
[tex]x+5=13\\x=8\\\\2x-3(5)=1\\2x-15=1\\2x=16\\x=8[/tex]
Answer:In short, the first number (x) is 8 and the second number (y) is 5.
How many cubes with an edge length of 1/3 inch are needed to build a cube with an edge length of 1 inch?
The number of cubes needed with an edge length of [tex]\bold{\dfrac{1}{3}}[/tex] inches is needed to build a cube with an edge length of 1 inch is 27.
Given to us,the edge length of smaller cube, a = [tex]\bold{\dfrac{1}{3}}[/tex] inches
the edge length of the cube to be built, S = [tex]\bold{\dfrac{1}{3}}[/tex] inches
Volume of a Cubewe know that volume of a cube is given by (side)³.
[tex]\bold{Volume\ of\ cube = (side)^3}[/tex]
Volume of the smaller cubeVolume of the smaller cube = (edge length of the smaller cube)³
= a³
= [tex]\bold{(\dfrac{1}{3})^3}[/tex]
= [tex]\bold{\dfrac{1}{27}}[/tex] in.³
Volume of the cube to be buildVolume of the cube to be build = (edge length of the cube to be built)³
= S³
= 1³
= 1 in.³
solving,
Cubes of smaller length are needed for the larger cube
[tex]\bold{=\dfrac{Volume\ of\ the\ Larger\ cube}{Volume\ of\ the\ Smaller\ cube}}[/tex]
[tex]\bold{=\dfrac{1}{\dfrac{1}{27}}}[/tex]
[tex]\bold{=27}[/tex]
Hence, the number of cubes needed with an edge length of [tex]\bold{\dfrac{1}{3}}[/tex] inches is needed to build a cube with an edge length of 1 inch is 27.
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The area of the square embedded in the circular base of the cylinder is 18 in^2. If the cylinder is 10 inches long, what is the surface area of the cylinder
The surface area of the cylinder is approximately 191.9 square inches.
Describe Surface Area of cylinder?The surface area of a cylinder is the sum of the areas of its curved surface and its two circular bases. The formula for the surface area of a cylinder is:
SA = 2πr² + 2πrh
where SA is the surface area, r is the radius of the circular base, h is the height of the cylinder, and π is the mathematical constant pi (approximately equal to 3.14159).
The first term on the right-hand side of the equation (2πr²) represents the area of the two circular bases of the cylinder. The second term (2πrh) represents the area of the curved surface of the cylinder.
The area of the square embedded in the circular base of the cylinder is 18 in², which means each side of the square has a length of √18 in = 3√2 in. Since the square is embedded in the circular base, the diameter of the base of the cylinder is equal to the diagonal of the square, which is 2 times the length of one side, or 6√2 in.
The radius of the circular base is half the diameter, or 3√2 in. The surface area of the cylinder consists of the area of the top circular base, the area of the bottom circular base, and the lateral area (the curved surface area) of the cylinder.
The area of each circular base is πr² = π(3√2)² = 18π in².
The lateral area of the cylinder can be found by multiplying the height (which is given as 10 inches) by the perimeter of the circular base (which is 2πr). The perimeter of the circular base is 2π(3√2) = 6π√2 in. Therefore, the lateral area is 10(6π√2) = 60π√2 in².
The total surface area of the cylinder is the sum of the areas of the two circular bases and the lateral area:
Surface area = 2(18π) + 60π√2
= 36π + 60π√2
≈ 191.9 in² (rounded to one decimal place)
Therefore, the surface area of the cylinder is approximately 191.9 square inches.
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Does (-2,0 ) make the equation y=x true ?
Answer:
False
Step-by-step explanation:
(-2,0) x is equal to -2 and y = 0
y=x
Substitute the values in.
-2 =0
False
What is the value of the expression? (−3)3+6312 Enter your answer in the box.
the value of the expression is approximately - 25.4349.
To find the value of the expression (- 3)³ + 6⁽³/¹²⁾, we follow the order of operations. First, we evaluate the exponentiation:
(- 3)³ = (- 3) × (- 3) × (- 3) = -27
Next, we simplify the second term:
6⁽³/¹²⁾ = 6⁽¹/⁴⁾ = √( √6) ≈ 1.5651
Finally, we add the two results:
(- 3)³ + 6⁽³/¹²⁾ = - 27 + 1.5651 ≈ - 25.4349
So, the value of the expression is approximately - 25.4349. The explanation highlights the steps of evaluating the exponentiation and simplifying the second term before performing the addition to get the final result.
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What is the slope of y = -2x?
Answer:
slope = - 2
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y-intercept )
y = - 2x is in this form
with slope m = - 2 and c = 0
A scale drawing of a square has side length of 2 inches the drawing has a scale is 1in:9mi find the actual perimeter and area of the square
The solution is: Area of actual Patio is: 170 ft²
Given:
Patio drawing of 4.25 in by 2.5 in
Scale of Patio to its drawing = 4ft to 1 in
Requires:
Area of actual Patio
SOLUTION:
Area of actual Patio = area of a rectangle = length × width
Given a scale drawing of 4ft to 1 in, therefore:
Length of actual Patio = 4.25 × 4 = 17 ft
Width of actual Patio = 2.5 × 4 = 10 ft
Therefore:
Area of actual Patio = 17 ft × 10 ft = 170 ft²
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complete question:
A contractor is given a scale drawing of a rectangular patio. The scale from the patio to the drawing is 4ft to 1in. What is the area of the actual patio?
The number of bacteria present in a culture after t minutes is given as b=1000e^kt where k is a constant. If there are 8520 bacteria present after 15 minutes, find k and round to the nearest thousandth. a. 0.143 c. 2.143 b. 32.136 d. 0.13
To solve this problem, plug 8250 in for b and 15 in for t and then solve for k:
[tex]8250=1000e^{15k}[/tex]
*Divide both sides by 1000*
[tex]8.25=e^{15k}[/tex]
*Take the natural log (ln) of both sides*
2.1102=15k
*Divide both sides by 15*
0.1406=k
0.1406 rounds to 0.141 which is close to a. 0.143.
Hope this helps!!
Can you help me on this?
Answer:
y = 30
Step-by-step explanation:
We can use ratio's to solve this
40 20
----- = -----------
100 20 + y
Using cross products
40 * (20+y) = 20*100
Distribute
800 + 40y = 2000
Subtract 800 from each side
40y = 2000-800
40y = 1200
Divide by 40
y = 1200/40
y =30
A juice company sells its product in either a 48-ounce size or a 32-ounce size. It charges $\$3.90$ for the 48-ounce size. How much should it charge for the smaller size if it wants the price per ounce to be $25\%$ more than the price per ounce of the larger size?
Answer:
The company should charge $ 3.25 for the smaller size.
Step-by-step explanation:
The price per ounce of the 32-ounce product is required to be 25% greater than the larger-sized product.
We know that for the size of 48 ounces the price is $ 3.90
So, the price per ounce is:
[tex]\frac{3.90}{48}[/tex]= $ 0.08125
For the smallest size the price per ounce should be 25% higher.
So:
$ 0.08125 (1 + 0.25) = $ 0.10156
The total price for the smallest size is:
$ 0.10156 * 32 = $ 3.25
The company should charge $ 3.25 for the smaller size.
Answer:
3.25
Step-by-step explanation:
We could solve this problem by figuring out the per-ounce cost of the 48-ounce package, increasing it by $25\%$, and then multiplying that by 32 for the smaller package. However, if we simply increase the price by $25\%$, and then scale the package size down to 32 ounces from 48 ounces, these are the same calculations, but in a different order that makes it easier to calculate. Thus: $3.90 \times 1.25 \times \frac{32}{48} = \boxed{3.25\text{ dollars}}$
What is the sixth term in the addition pattern that begins with 15,29,43,57?
A)61
B)71
C)85
D)99
Answer:
C is your answer
Step-by-step explanation:
Add every number by 14
If the Greatest Common Factor of L and M is 6, write the expression for the Least Common Multiple of these numbers.
Answer:
[tex]\frac{(L x M)}{6}[/tex]
Step-by-step explanation:
The greatest common factor is the greatest number that will divide two values. We have two values L and M. Each has numbers which multiply together to give the number. The highest value or most in common they share is 6. This is the GCF.
The least common multiple is the smallest positive number which is a multiple of the two. This means both L and M divide into it evenly.
We know L x M is a multiple because L and M will be factors of it. But we don't know its the least.
As an example if L= 42 and M = 60, they have GCF 6. We can multiply them to find a multiple 42 x 60 = 2520 but we don't know this is the smallest or least multiple we can find. If we divide by the GCF, 2520/6=420. Interestingly, 42 x 10 =420 and 60 x 7 =420. This means 420 is the least common multiple.
We can multiply (L x M) and then divide by the GCF of L & M to find the least common multiple.
[tex]\frac{(L x M)}{6}[/tex]