The optimal values of (a,b) that minimize the error function f(a,b) are approximately (0.7205, 0.5369).
What is a function?
A function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output.
To find the optimal values of (a,b), we need to minimize the error function f(a,b). We can do this by taking partial derivatives of f(a,b) with respect to both a and b, and then setting them equal to zero:
∂f/∂a = 2∑([tex]x_{i}^{2}[/tex])(a [tex]x_{i}^{2}[/tex]+ b [tex]y_{i}^{2}[/tex] - 1) = 0
∂f/∂b = 2∑([tex]y_{i}^{2}[/tex])(a [tex]x_{i}^{2}[/tex] + b [tex]y_{i}^{2}[/tex] - 1) = 0
We can simplify these equations by defining the following sums:
Sxx = ∑[tex]x_{i}^{4}[/tex]
Syy = ∑[tex]y_{i}^{4}[/tex]
Sxy = ∑[tex]x_{i}^{2}y_{i}^{2}[/tex]
Sx = ∑[tex]x_{i}^{2}[/tex]
Sy = ∑[tex]y_{i}^{2}[/tex]
Using these sums, we can rewrite the partial derivatives as:
∂f/∂a = 2(aSx² + bSxy² - Sx)
∂f/∂b = 2(aSxy² + bSy² - Sy)
Setting these equal to zero and solving for a and b, we get:
a = (SySx - Sxy²) / (SxSyy - Sxy²)
b = (SxSy - Sxy²) / (SxSyy - Sxy²)
Plugging in the values for Sxx, Syy, Sxy, Sx, and Sy, we get:
a = 0.7205
b = 0.5369
Therefore, the optimal values of (a,b) that minimize the error function f(a,b) are approximately (0.7205, 0.5369).
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A plane is flying at a speed of 320 miles per hour on a bearing of N65°E. Its ground speed is 390 miles per hour and its true course, given by the direction angle of the ground speed vector, is 30°. Find the speed, in miles per hour, and the direction angle, in degrees, of the wind.
The speed in miles per hour is 111.2 and the direction angle in degrees is 260.2.
We are given the speed of a plane on a bearing of N [tex]75^\circ[/tex] E and its ground speed. We have to find its speed in miles per hour and the direction angle in degrees. We will apply the formula of projection for both the x-axis and y-axis.
As we know, projection, R = V + W
Now, the x-axis projection will be R cos[tex]15^\circ[/tex] according to the angle given to us. Therefore, R cos [tex]15^\circ[/tex] = V cos[tex]30^\circ[/tex] + [tex]W_{x}[/tex]
The y-axis projection,
R sin [tex]15^\circ[/tex] = V sin [tex]30^\circ[/tex] + [tex]W_{y}[/tex]
From here, now we will find [tex]W_{x}{[/tex] and [tex]W_{y}[/tex]
[tex]W_{x}{[/tex] = 330 cos[tex]15^\circ[/tex] - 390 cos[tex]30^\circ[/tex]
[tex]W_{x}[/tex] = -19 miles/hour
[tex]W_{y}[/tex] = 330 sin[tex]15^\circ[/tex] - 390 sin[tex]30^\circ[/tex]
[tex]W_{y}{[/tex] = -109.6 miles per hour
Now, W = [tex]\sqrt{(W_{x})^{2} + (W_{y})^{2{}}[/tex]
W = [tex]\sqrt{(-19.0)^{2} + (-109.6})^{2{}}[/tex]
W = 111.2 miles/hour
Now, we will find the angle with the help of tan θ.
tan θ = [tex]\frac{W_{y}}{W_{x}}[/tex]
tan θ = [tex]\frac{-109.6}{-19.0}[/tex]
θ = [tex]tan ^{-1} (\frac{109.6}{19.0})[/tex]
θ = 260.[tex]2^\circ[/tex]
Therefore, the speed in miles per hour is 111.2 and the direction angle in degrees is 260.2.
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PLEASE HURRY
Write the vector v in terms of i and j whose magnitude and direction angle are given ||v|| = 2/3, theta = 116 deg
The vector v can be expressed as v = -0.161 i + 0.618 j, where i and j are the unit vectors
To express a vector v in terms of i and j, we need to find its x and y components. The magnitude ||v|| of a vector is given by:
||v|| = √(v₁² + v₂²)
where v₁ and v₂ are the x and y components of v, respectively.
The direction angle θ of a vector with respect to the positive x-axis is given by:
θ = atan(v₂/v₁)
where atan denotes the arctangent function.
In this problem, we are given that the magnitude ||v|| of the vector v is 2/3, and its direction angle θ with respect to the positive x-axis is 116 degrees. Therefore, we can write:
||v|| = √(v₁² + v₂²) = 2/3
θ = atan(v₂/v₁) = 116°
Solving for the x and y components, we get:
v₁ = ||v|| cos(θ) = (2/3) cos(116°) ≈ -0.161
v₂ = ||v|| sin(θ) = (2/3) sin(116°) ≈ 0.618
Therefore, the vector v can be expressed as:
v = -0.161 i + 0.618 j
where i and j are the unit vectors in the x and y directions, respectively.
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Please help me with this math problem
1. Josiah is making a candle by pouring melted wax into a mold in the shape of a square pyramid. Each side of the base of the pyramid is 12 in and the height of the pyramid is 14in. To get the wax for the candle, Josiah melts cubes of wax that are each 6 in by 6 in by 6 in. How many of the wax cubes will Josiah need in order to make the candle? Show your work.
Answer:
10 wax cubes
Step-by-step explanation:
the volume of the square pyramid can be found by using the following formula:
[tex]V = \frac{1}{3} * B * h[/tex]
where b is the base of the pyramid and h is the height of the pyramid
The area of the base of the pyramid is given as 12 in * 12 in = 144 in^2
So, the volume of the pyramid is:
V = (1/3) * 144 in^2 * 14 in = 2,016 in^3
Each wax cube has a volume of 6 in * 6 in * 6 in = 216 in^3.
To find the number of wax cubes needed, we can divide the volume of the pyramid by the volume of each wax cube:
Number of wax cubes = Volume of pyramid / Volume of each wax cube
Number of wax cubes = 2,016 in^3 / 216 in^3
Number of wax cubes = 9.33 (rounded to two decimal places)
Since we can't have a fraction of a wax cube, Josiah will need to use 10 wax cubes to make the candle.
3. Abby has 3 bags of oranges with 14,
23 and 28 oranges in them. Abby
rounds the number of oranges in each
bag to the nearest ten. About how
many oranges does Abby have
altogether?
Answer:
60
Step-by-step explanation:
14 rounds to 10
23 rounds to 20
28 rounds to 30
10+20+30=60
2010 2008
$971 $812
$977 $943
$900 $873
$1071 $1023
$501 $486
Average Weekly Earnings in Canada
Occupation
Forestry, logging and support
Manufacturing
Transportation and warehousing
Construction
Retail trade
1. Calculate the mean (average) weekly earnings of workers in the occupations
listed for 2010.
Answer:
Step-by-step explanation:
To calculate the mean (average) weekly earnings of workers in the occupations listed for 2010, we need to add up the earnings for each occupation and divide by the total number of occupations.
Forestry, logging and support: $971
Manufacturing: $977
Transportation and warehousing: $900
Construction: $1071
Retail trade: $501
Total earnings: $4,420
Total number of occupations: 5
Mean weekly earnings: $4,420 ÷ 5 = $884
Therefore, the mean (average) weekly earnings of workers in the occupations listed for 2010 is $884.
Question 10 of 10
A minor arc will have a measure that is
O A. less than 180°
B. equal to 180°
OC. more than 180°
Answer:
Less than 180
Step-by-step explanation:
Minor: Less than 180
Major: More than 180
PLEASE HELP ME WITH THE WHOLE PROBLEM PLEASEEEE
The probability of getting blue or green is 1/3, the probability of getting blue is 3/4 and there are 8 blue marbles.
What is the probability in each case?a) If the probability of getting a yellow marble is 2/3, then the probability of getting a blue or green marble is:
P(blue or green) = 1 - P(yellow) = 1 - 2/3 = 1/3
b) If the probability of getting a green marble is 1/4, then the probability of getting a blue marble is:
P(blue) = 1 - P(green) = 1 - 1/4 = 3/4
c) The total number of marbles in the bag is 24. Let x be the number of blue marbles. Then the number of yellow marbles is 2x, and the number of green marbles is 24 - x - 2x = 24 - 3x. We know that:
2x/24 = 2/3 (probability of getting a yellow marble is 2/3)
=> x = 8
Therefore, there are 8 blue marbles in the bag.
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lim h -> 0 [f(x_{0} + h) - f(x_{0})] / h
the limit expression gives the value of the derivative of a function at a specific point. where f'(x_0) denotes the derivative of f(x) at x = x_0.
what is derivative ?
The derivative of a function is a measure of how the function changes as its input variable changes. It gives the instantaneous rate of change or slope of the tangent line of the function at a specific point.
In the given question,
The expression you provided represents the limit definition of the derivative of a function f(x) at the point x = x_0. The limit evaluates the instantaneous rate of change or slope of the tangent line of the function f(x) at the point x = x_0.
To evaluate the limit, substitute x = x_0 + h in the expression of the function f(x) and simplify:
[tex]lim h - > 0 [f(x_{0} + h) - f(x_{0})] / h = f'(x_{0})[/tex]
where f'(x_0) denotes the derivative of f(x) at x = x_0.
Therefore, the limit expression gives the value of the derivative of a function at a specific point.
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A store sells rectangular picture frames in two sizes. The shorter side of the larger picture frame is 8 inches long and its longer side is 10 inches long. The longer side of the smaller picture frame is 6 inches long. The picture frames are similar shapes. What is the length of the shorter side of the smaller picture frame? Enter your answer as a decimal in the box.
inches
Answer: 4.8 Inches
Step-by-step explanation:
6 is 60% of 10
Therefore (60%*8 = 4.8)
*since they are similar, and therefore proportional
One factor of this polynomial is (x + 8).
2) + 522 - 11x + 104
Use synthetic division to find the other factor of the polynomial.
Answer:
We can use synthetic division to divide the polynomial 2x^3 + 522x^2 - 11x + 104 by x + 8, since x + 8 is a factor of the polynomial:
-8 | 2 522 -11 104
| -16 -412 2584
|--------------------
2 506 -423 2688
The numbers in the bottom row of the synthetic division represent the coefficients of the quotient polynomial, in order of decreasing degree. So the quotient polynomial is:
2x^2 + 506x - 423
Therefore, the other factor of the polynomial is 2x^2 + 506x - 423.
50 Points! Multiple choice algebra question. Photo attached. Thank you!
Answer: A
Step-by-step explanation:
We can use the logarithmic identity logb(a^n) = n*logb(a) to solve this problem.
First, we need to express 4 as a power of 2. We know that 2^2 = 4, so we can write:
log4 = log2^2
Then we can use the identity to rewrite this as:
log4 = 2*log2
Now we can use the given approximation log2 ≈ 0.4307 to approximate log4:
log4 ≈ 2 * 0.4307
log4 ≈ 0.8614
Therefore, the answer is (A) 0.8614.
Use the even and odd properties of the following functions to answer the questions:
a) If secθ=-3.1,then sec(-θ)=?
b)If sinθ=0.62,then sin(-θ)=?
Answer:
[tex]sec(-\theta)=-3.1[/tex]
[tex]sin(-\theta)=-0.62[/tex]
Step-by-step explanation:
Part a.
[tex]sec(\theta)[/tex] is related to [tex]cos(\theta)[/tex] through a reciprocal relationship [tex]sec(\theta)=\frac{1}{cos(\theta)}[/tex].
Since [tex]cos(\theta)[/tex] is an even function (reflecting left-right across the y-axis doesn't change the graph), then [tex]sec(\theta)[/tex] is also an even function.
For any input in the domain, even functions produce the same output if the opposite of the input is used. In other words, if "f" is an even function, for all x in the domain of f, [tex]f(x)=f(-x)[/tex].
Thus, for the secant function, if a mystery value "x" is used as an input, and -3.1 is obtained as an output, then if the opposite of x, or -x, is input into the secant function, the output will also be -3.1.
[tex]sec(-\theta)=-3.1[/tex]
Part b.
The [tex]sin(\theta)[/tex] function does not reflect left-right across the y-axis to produce the same graph, so the sine function is not even.
However, the sine function can be rotated 180 degree about the origin, sometimes thought of as reflecting through the origin, to produce the same graph. Visually, this is an "odd" function.
For any input in the domain, if the opposite of the input is used, odd functions produce the opposite of the original output. In other words, if "g" is an odd function, for all x in the domain of g, [tex]-g(x)=g(-x)[/tex].
Thus, for the sine function, if a mystery value "x" is used as an input, and 0.62 is obtained as an output, then if the opposite of x, or -x, is input into the sine function, the output will be the opposite of 0.62, meaning -0.62.
[tex]sin(-\theta)=-0.62[/tex]
Find m so that x + 5 is a factor of - 3x^4 - 10x^3 + 20x^2 - 22x + m.
If x + 5 is a factor of the given polynomial, then (x + 5) must divide the polynomial evenly, meaning that the remainder is 0 when the polynomial is divided by x + 5.
We can use polynomial long division or synthetic division to find the quotient and remainder, but it's easier to use the fact that if x + 5 is a factor, then (-5) must be a root of the polynomial.
So, we can substitute x = -5 into the polynomial and set it equal to 0 to find m:
-3(-5)^4 - 10(-5)^3 + 20(-5)^2 - 22(-5) + m = 0
Simplifying and solving for m:
-3(625) + 10(125) + 20(25) + 110 + m = 0
-1875 + 1250 + 500 + 110 + m = 0
m = 1015
Therefore, m = 1015 so that x + 5 is a factor of - 3x^4 - 10x^3 + 20x^2 - 22x + m.
Convert the decimal 15.75 into a percent.
The decimal 15.75 into a percentage is 1575%
How to convert the decimal into a percentageFrom the question, we have the following parameters that can be used in our computation:
Decimal = 15.75
This means that
Number = 15.75
Multiply the number by 1
so, we have the following representation
Number = 15.75 * 1
Express 1 as 100%
This gives
Number = 15.75 * 100%
Evaluate the products of 15.75 and 100
So, we have
Number = 1575%
Hence, the percentage is 1575%
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College Level Trig Question Any help will do!!
The value of x in the equation y = 9 sec(2x) at [0, π/4) ∪ (π/4, π/2] is x = 1/2[sec₋¹(y/9)]
Calculating the values of xFrom the question, we have the following parameters that can be used in our computation:
y = 9 sec(2x)
The interval of x is also given as
[0, π/4) ∪ (π/4, π/2]
This means that the values of x is from 0 to π/2, however, the function is undefined at x = π/4 i.e. there is a hole at x = π/4
Next, we set the equation to y
So, we have
9 sec(2x) = y
Divide both sides by 9
sec(2x) = y/9
Take the arc sec of both sides of the equation
2x = sec₋¹(y/9)
Divide both sides by 9
x = 1/2[sec₋¹(y/9)]
Hence, the value of x is x = 1/2[sec₋¹(y/9)]
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A red marble is drawn from a bag containing 3 red and 3 blue marbles. If the red marble is not replaced, find the probability of drawing a second marble that is blue.
The probability of drawing a second marble that is blue is 3/5
Finding the probability of drawing a second marble that is blue.From the question, we have the following parameters that can be used in our computation:
A red marble is drawn from a bag containing 3 red and 3 blue marbles.
If the marbles were not replaced, then we have
P(Red) = 3/6
Now there are
3 blue marbles and 2 red marbles left
So, we have
The probability of choosing a blue marble, after a red marble is
P(Blue) = 3/5
Evaluate
P(Blue) = 3/5
Hence, the probability of choosing a blue marble, after a red marble is 3/5
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Evaluate.
(49)−2⋅(34)−2
Answer: To evaluate this expression, we need to follow the order of operations, which is PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction) and work from left to right:
(49)^(-2) * (34)^(-2)
First, we can simplify the exponents:
1/(49^2) * 1/(34^2)
Next, we can calculate the values of 49^2 and 34^2:
1/2401 * 1/1156
Then, we can multiply the fractions:
1/2785456
Therefore, the value of the expression (49)^(-2) * (34)^(-2) is approximately 0.000000359.
Step-by-step explanation:
A group of 7 friends is planning a hike. Each friend will need of a gallon of water to drink during the hike. How many gallons of water will the group need for the hike?
Use the unit circle to find the exact value of the trig function
cos(210°)
Answer:
-[tex]\sqrt{3}[/tex]/2
Step-by-step explanation:
cos is negative in quad II
cos(210)= -cos(30) = -[tex]\sqrt{3}[/tex]/2
Please help me out it’s a new topic and I don’t know how to do it
Answer:
=x2-100
Step-by-step explanation:
When I said x2 I mean (x*x)
what are the answers to these questions?
If the line passes through the point (2,8) that cuts off the least area from the first quadrant, the slope is 8/3 and the y-intercept is 0.
To find the equation of the line that passes through the point (2, 8) and cuts off the least area from the first quadrant, we need to first determine the slope of the line. The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.
We can use the point-slope form of the equation of a line to find the slope. The point-slope form is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line. Plugging in (2, 8) as the point, we get:
y - 8 = m(x - 2)
Next, we want to minimize the area that the line cuts off in the first quadrant. Since the line passes through the origin (0, 0), the area cut off by the line in the first quadrant is equal to the product of the x- and y-intercepts of the line.
We can express the x-intercept in terms of y by setting y = 0 in the equation of the line and solving for x:
0 - 8 = m(x - 2)
x = 2 + 8/m
The y-intercept is simply the y-coordinate of the point where the line intersects the y-axis, which is given by:
y = mx + b
8 = 2m + b
b = 8 - 2m
We can now express the area cut off by the line as:
A = x*y
A = (2 + 8/m)*8 - (8 - 2m)*2/m
A = (16 + 64/m) - (16 - 4m)/m
A = 64/m + 4m/m
To minimize the area, we can take the derivative of A with respect to m and set it equal to zero:
dA/dm = -64/m² + 4/m² = 0
64 = 4
m = 8
Plugging m = 8 into the equation for the x-intercept, we get:
x = 2 + 8/8 = 3
So the equation of the line is y = 8x/3.
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22. The expression 3(-4b) - 2(a - b - c) is equal to which of the following expressions?
(1) -2a - 10b - 2c
(2) -2a - 10b + 2c
(3) -2a -5b + 2c
(4) -2a -4b - 2c
(5) 2a-4b - 2c
Answer:
We can simplify the expression as follows:
3(-4b) - 2(a - b - c) = -12b - 2a + 2b + 2c
Combining like terms, we get:
= -2a - 10b + 2c
Therefore, the expression 3(-4b) - 2(a - b - c) is equal to option (2), -2a - 10b + 2c.
Step-by-step explanation:
Im smart
One number is 8 more than another number, and their sum is 14. Find the numbers.
Answer: x=11, y=3
Step-by-step explanation:
x=y+8, name one x and one y, so x is 8 more than y
x+y=14, x plus y is 14
substitute x into second eqution, so (y+8)+y=14
simplify so 2y+8=14
y=3 and use that to find x
x=11
50 Points! Multiple choice algebra question. Which function represents exponential growth? Photo attached. Thank you!
The exponential growth model is represented by the function y = 10 · 3ˣ. (Correct choice: D)
What function is an exponential growth function?
In this problem we must determine what function does represent an exponential growth model. With this purpose, we must define and understand the following functions:
Exponential growth model
y = a · rˣ, for r > 1.
Exponential decay model
y = a · rˣ, for 0 < r < 1.
Polynomic model
y = ∑ cₙ · xⁿ
The function y = 10 · 3ˣ represents an exponential growth model.
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Michael invests $1,000 in an account that earns a 4.75% annual percentage rate compounded continuously. Peter invests$1,200 in an account that earns a 4.25% annual percentage rate compounded continuously. Which person's account will grow to $1,800 first?
Answer:
Step-by-step explanation:
We can use the formula for continuous compounding to determine how long it will take each account to reach $1,800.
For Michael's account, the formula is:
A = P*e^(rt)
where:
A is the amount in the account after t years
P is the principal
r is the annual interest rate
t is the time in years
Plugging in the values given, we get:
1,800 = 1,000*e^(0.0475t)
Taking the natural logarithm of both sides, we get:
ln(1,800/1,000) = 0.0475t
t = ln(1,800/1,000)/0.0475
t ≈ 8.55 years
So it will take approximately 8.55 years for Michael's account to reach $1,800.
Similarly, for Peter's account, the formula is:
A = P*e^(rt)
where:
A is the amount in the account after t years
P is the principal
r is the annual interest rate
t is the time in years
Plugging in the values given, we get:
1,800 = 1,200*e^(0.0425t)
Taking the natural logarithm of both sides, we get:
ln(1,800/1,200) = 0.0425t
t = ln(1,800/1,200)/0.0425
t ≈ 9.03 years
So it will take approximately 9.03 years for Peter's account to reach $1,800.
Therefore, Michael's account will grow to $1,800 first as it will take less time (8.55 years) compared to Peter's account (9.03 years).
Assume that women's heights are normally distributed with a mean given by μ = 63.6 in, and a standard deviation given by a = 2.5 in. Complete parts a and b.
a. If 1 woman is randomly selected, find the probability that her height is between 62.9 in and 63.9 in.
The probability is approximately
(Round to four decimal places as needed.)
The probability that the woman's height is between 62.9 in and 63.9 in is 0.1580
Calculating the probability of values from the the z-scoresFrom the question, we have the following parameters that can be used in our computation:
Mean = 63.6Standard deviation = 2.5Age = between 62.9 and 63.9So, the z-scores are
z = (62.9 - 63.6)/2.5 = -0.28
z = (63.9 - 63.6)/2.5 = 0.12
i.e. between a z-score of -0.28 and a z-score of 0.12
This is represented as
Probability = (-0.28 < z < 0.12)
Using a graphing calculator, we have
Probability = 0.1580
Hence, the probability is 0.1580
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The function f(x) = 2-5* can be used to represent the curve through the points (1, 10), (2, 50), and (3, 250). What is the multiplicative rate of change of the function? 0 2 05 10 • 32
The given function does not represent the curve passing through the given points. The multiplicative rate of change of the function is 5.
What is Curve Line ?A curve is a continuous, smooth line that gradually alters direction. Rather than being straight line that follows a curve may be referred to as a curve line. A curve line can be made by connecting a number of non-straight-line-lying sites. In mathematics, a curve is a geometrical object that can be described by equations, such as parametric or function equations.
The rate of change of a function is the ratio of the change in the output (y) to the change in the input (x). In this case, we can calculate the rate of change between the first two points:
Changing at what rate between (1, 10) and (2, 50)?
Y change = 50 – 10 = 40
Variation in x = 2 - 1 = 1
Rate of change equals change in y/change in x, or 40/1, or 40.
In a similar manner, we can determine how quickly the second and third points will change:
The change between (2, 50) and (3, 250) is as follows:
Changing y by 250 - 50 equals 200
Variation in x = 3 - 2 = 1
Rate of change equals change in y/change in x, which is 200/1, or 200.
Now that we have the second rate of change, we can compute the multiplicative rate of change by dividing it by the first:
Multiplicative rate of change is equal to 200/40, or 5.
The function's multiplicative rate of change is thus 5.
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2. How does making tables help you
identify relationships between terms in
patterns?
Answer:
Step-by-step explanation:
well if you know the term than you know the pattern
Solve and check: 3/4h+11=20 helpp!
Answer: 12
Step-by-step explanation:
We have to find h so the first step is to subtract the 11 from the 20
3/4h=9
Then to move the 3/4 over to the other side, you must multiply it by it's reciprocal, 4/3.
h= 4/3 x 9
This equals 12.
find two numbers who’s product is -9 and who’s sum is -8
Answer:
1, -9
Step-by-step explanation:
xy = -9
x + y = -8 solve this for x and substitute into the 1st equation
x = -8 - y
(-8 - y)y = -9
-y² - 8y + 9 = 0
y² + 8y -9 = 0
Solve for y by factoring:
(y - 1)(y + 9) = 0
y = 1, -9
x(1) = -9
x = -9
x(-9) = -9
x = -9/-9 = 1
