a. The minimum-cost healthy purchase snack problem can be formulated as a linear optimization problem with the objective to minimize cost. The decision variables are x_b for the number of brownies purchased and x_c for the number of mini-cheesecakes purchased. The constraints are as follows:
The linear optimization problem can be solved using the linprog routine from the scipy.optimize library.
b. The Lagrangian for this problem can be written as follows:
L = 50x_b + 80x_c + λ_1(3x_b + 0x_c - 6) + λ_2(2x_b + 4x_c - 10) + λ_3(2x_b + 5x_c - 8),
where λ_1, λ_2 and λ_3 are the Lagrange multipliers associated with the constraints.
The optimality conditions for this problem are:
c. The values of the Lagrange multipliers can be computed by solving the system of equations given by the optimality conditions. Their physical interpretation in this problem is that they are proportional to the amount by which the constraints are violated. Positive values of the Lagrange multipliers indicate that the corresponding constraints are active, while negative values indicate that the constraints are inactive. The values of the Lagrange multipliers will depend on the optimal solution.
d. The problem can be converted to canonical form by rewriting the inequality constraints as:
The canonical form of the problem is used internally in the solution.
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Using long division to find each quotient
(2x³ + x²-x-4) ÷ (x + 4)
Answer:
The quotient of (2x³ + x² - x - 4) ÷ (x + 4) is 2x^2 - 7x + 17, and the remainder is 27x - 4.
Step-by-step explanation:
2x^2 - 7x + 17
x + 4 | 2x^3 + x^2 - x - 4
- (2x^3 + 8x^2)
---------------
-7x^2 - x
+ (-7x^2 - 28x)
-------------
27x - 4
Therefore, the quotient of (2x³ + x² - x - 4) ÷ (x + 4) is 2x^2 - 7x + 17, and the remainder is 27x - 4.
help me do this pls it is to hard
The fraction amount of pizza eaten is given by A = ( 5/6 ) of the pizza
The amount of pizza left over B = ( 1/6 ) of the pizza
What is a Fraction?An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator or denominator of a complex fraction. The numerator of a proper fraction is less than the denominator.
Given data ,
Let the fraction of pizza eaten be represented as A
Now , the value of A is
Let the amount of pizza eaten by Tyler be p = ( 1/2 ) of the pizza
Let the amount of pizza eaten by Dean be q = ( 1/3 ) of the pizza
Now , the fraction of pizza eaten A = p + q
On simplifying , we get
The fraction of pizza eaten A = ( 1/2 ) + ( 13 )
The fraction of pizza eaten A = ( 3 + 2 ) / 6
The fraction of pizza eaten A = ( 5/6 ) of the pizza
And , the amount of pizza left B = 1 - A
On simplifying , we get
The amount of pizza left = 1/6 of the total pizza
Hence , the fractions are solved
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what is the difference between inverse and direct proportions?
Answer:
Direct proportion and inverse proportion are two types of relationships between two variables.
Direct proportion is a relationship in which two variables increase or decrease together at the same rate. In other words, if one variable increases, the other variable also increases, and if one variable decreases, the other variable also decreases. The mathematical expression for direct proportion is:
y = kx
where y is the dependent variable, x is the independent variable, and k is a constant of proportionality.
On the other hand, inverse proportion is a relationship in which two variables change in opposite directions. In other words, if one variable increases, the other variable decreases, and if one variable decreases, the other variable increases. The mathematical expression for inverse proportion is:
y = k/x
where y is the dependent variable, x is the independent variable, and k is a constant of proportionality.
So, the main difference between inverse proportion and direct proportion is the direction of change between the two variables. In direct proportion, the two variables change in the same direction, while in inverse proportion, the two variables change in opposite directions.
How did this happen? Hannah deposits $630 in a savings account at 2. 75% simple interest
Hannah will have $681.98 in her savings account after 3 years at 2.75% simple annual interest.
To calculate the amount of money Hannah will have in the account after 3 years, we can use the formula for simple interest:
Simple interest = principal x rate x time
where:
principal is the amount of money initially deposited
rate is the annual interest rate (expressed as a decimal)
time is the number of years
In this case, the principal is $630, the rate is 2.75% (or 0.0275 as a decimal), and the time is 3 years. Plugging these values into the formula, we get:
Simple interest = $630 x 0.0275 x 3 = $51.98
This means that after 3 years, Hannah will have earned $51.98 in simple interest. To find the total amount of money in her account, we need to add this interest to the original principal:
Total amount = principal + simple interest
Total amount = $630 + $51.98 = $681.98
Therefore, after 3 years Hannah will be having $681.98 in her account after 3 years at 2.75% simple annual interest.
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The complete question is :
Hannah deposits $630 in a savings account at 2. 75% simple annual interest. How much money will she have in the account after 3 years?
Alexandri downloads 13 songs that cost 0.99 each plus song for 1.49.She uses a coupon for 1.50 off.What is the total price Alexandria pays?
So, by resolving the given, we obtain the result: Alexandria thus shells expressions out a total of $12.86 for the 14 tracks.
what is expression ?Mathematical operations include doubling, dividing, adding, and subtracting. A phrase is constructed as follows: Expression, monetary value, and mathematical operation Numbers, parameters, and functions make up a mathematical expression. It is possible to use words and terms in contrast. Every mathematical statement including variables, numbers, and a mathematical operation between them is called an expression, sometimes referred to as an algebraic expression. For example, the expression 4m + 5 is composed of the expressions 4m and 5, as well as the variable m from the above equation, which are all separated by the mathematical symbol +.
Alexandria downloads 13 tracks at $0.99 each, for a grand total of $12.87 (13 x $0.99).
She also spends $1.49 downloading one song.
The price before the coupon is thus $12.87 + $1.49 = $14.36.
The final price is $14.36 - $1.50 = $12.86 after the coupon for $1.50 discount is applied.
Alexandria thus shells out a total of $12.86 for the 14 tracks.
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Choose all the expressions that are equal to 1/6. A. 6÷1 B. 3÷18 C. 2÷ 1/3 D. 1÷6 E. 1/3 ÷ 2
We can state this by responding to the provided question B, D, and E are expressions the expressions that have values of 1/6.
what is expression ?Mathematical operations include doubling, dividing, adding, and subtracting. A phrase is constructed as follows: Expression, monetary value, and mathematical operation Numbers, parameters, and functions make up a mathematical expression. It is possible to use words and terms in contrast. Every mathematical statement including variables, numbers, and a mathematical operation between them is called an expression, sometimes referred to as an algebraic expression.
We need to simplify each expression to see which ones are equivalent to 1/6:
6. 1/6 is not equivalent to 6/1.
B. 3÷18 = 1/6
C. 2/3 of 2 equals 6 (not equivalent to 1/6)
D. 1÷6 = 1/6
E. 1/3 ÷ 2 = 1/6
B, D, and E are the expressions that have values of 1/6.
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Consider the code over F4 with generator matrix 11 10 01 10 00
11 Construct a standard array for the code, and use it to decode
the words 00 11 01 and 11 11 10.
To decode the words 0011, 0101, 0110 and 1111, we can find the corresponding codeword in the standard array. The codewords for 0011, 0101, 0110 and 1111 are 0011, 0101, 0110 and 1111, respectively.
To construct a standard array for this code, we can start by finding the generator polynomial of the code. This can be done by multiplying the generator matrix with its transpose and finding the resultant matrix. This results in the generator polynomial: 110101001.
To construct the standard array, we first construct a parity check matrix, which is the transpose of the generator matrix: 0101 1001 1100 1000.
From the parity check matrix, we can then construct the standard array. The standard array consists of all the possible codewords given the generator polynomial and the parity check matrix. It can be constructed by multiplying the generator matrix with all possible vectors of length 3.
The standard array of the code is:
0011 0101 0110 1001 1010 1100 1101 1111
To decode the words 0011, 0101, 0110 and 1111, we can find the corresponding codeword in the standard array. The codewords for 0011, 0101, 0110 and 1111 are 0011, 0101, 0110 and 1111, respectively.
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(URGENT!) Please show work as well.
Answer:
Step-by-step explanation:
[tex]-\frac{1}{2}[/tex] x ≤ 17
( [tex]-\frac{2}{1}[/tex] )( [tex]-\frac{1}{2}[/tex] ) x ≤ 17( [tex]-\frac{2}{1}[/tex] )
x ≥ - 34
(a) Show that the F statistic for testing that all effects equal
0 has expression in terms of the R2 value as
(b) Show that the F statistic for comparing nested models has
expression in terms of the
(a) The F statistic for testing that all effects equal 0 is given by:
F = [(R2 / k) / ((1 - R2) / (n - k - 1))]
where R2 is the coefficient of determination, k is the number of predictor variables, and n is the total number of observations.
To derive this expression, we start with the definition of the F statistic:
F = [(SSR / k) / (SSE / (n - k - 1))]
where SSR is the sum of squares for regression and SSE is the sum of squares for error.
We know that R2 = SSR / SST, where SST is the total sum of squares. Therefore, we can rewrite SSR as:
SSR = R2 * SST
Substituting this back into the F statistic, we get:
F = [(R2 * SST / k) / (SSE / (n - k - 1))]
We also know that SST = SSR + SSE, so we can substitute this back into the F statistic:
F = [(R2 * (SSR + SSE) / k) / (SSE / (n - k - 1))]
Simplifying the expression, we get:
F = [(R2 / k) / ((1 - R2) / (n - k - 1))]
This is the expression for the F statistic in terms of the R2 value.
(b) The F statistic for comparing nested models is given by:
F = [(R2_2 - R2_1) / (k_2 - k_1)] / [(1 - R2_2) / (n - k_2 - 1)]
where R2_1 and R2_2 are the R2 values for the smaller and larger models, respectively, and k_1 and k_2 are the number of predictor variables in the smaller and larger models, respectively.
To derive this expression, we start with the definition of the F statistic:
F = [(SSR_2 - SSR_1) / (k_2 - k_1)] / [(SSE_2 / (n - k_2 - 1))]
where SSR_1 and SSR_2 are the sum of squares for regression for the smaller and larger models, respectively, and SSE_2 is the sum of squares for error for the larger model.
We know that R2 = SSR / SST, so we can rewrite SSR_1 and SSR_2 as:
SSR_1 = R2_1 * SST
SSR_2 = R2_2 * SST
Substituting these back into the F statistic, we get:
F = [((R2_2 * SST) - (R2_1 * SST)) / (k_2 - k_1)] / [(SSE_2 / (n - k_2 - 1))]
Simplifying the expression, we get:
F = [(R2_2 - R2_1) / (k_2 - k_1)] / [(1 - R2_2) / (n - k_2 - 1)]
This is the expression for the F statistic for comparing nested models in terms of the R2 values.
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Find
f ∘ g
and
g ∘ f.
f(x) = x3, g(x) = x2/3
(a)
f ∘ g
(b)
g ∘ f
Find the domain of each function and each composite function. (Enter your answers using interval notation.)
domain of f domain of g domain of f ∘ g domain of g ∘ f
The composite functions f ∘ g and g ∘ f are:
domain of f = (-∞, ∞)
domain of g = (-∞, ∞)
domain of f ∘ g = (-∞, ∞)
domain of g ∘ f = (-∞, ∞)
The composite functions f ∘ g and g ∘ f are formed by plugging one function into the other. To find f ∘ g, we plug the function g into the function f:
f ∘ g = f(g(x)) = f(x2/3) = (x2/3)3 = x2
To find g ∘ f, we plug the function f into the function g:
g ∘ f = g(f(x)) = g(x3) = (x3)2/3 = x2
The domain of a function is the set of all possible values of x that make the function defined. The domain of f is all real numbers, since the function f(x) = x3 is defined for all values of x:
domain of f = (-∞, ∞)
The domain of g is also all real numbers, since the function g(x) = x2/3 is defined for all values of x:
domain of g = (-∞, ∞)
The domain of f ∘ g is the intersection of the domain of f and the domain of g, which is all real numbers:
domain of f ∘ g = (-∞, ∞)
The domain of g ∘ f is also the intersection of the domain of g and the domain of f, which is all real numbers:
domain of g ∘ f = (-∞, ∞)
Therefore, the answers are:
domain of f = (-∞, ∞)
domain of g = (-∞, ∞)
domain of f ∘ g = (-∞, ∞)
domain of g ∘ f = (-∞, ∞)
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Johnny is the Senior Class President and wants to buy Senior Class T-Shirts. The T-
Shirt company is offering the following deal:
-$25 per shirt if you buy 40 shirts or less
- $20 per shirt if you buy more than 40 shirts and up to 100 shirts
-$15 per shirt if you buy more than 100 shirts
Create a Piecewise Function f(x) that you can use to determine the total price of the
shirts. Let x be the number of shirts you order. Use the inequality buttons below to
help you input the Domains for B,D, and F.
A.
B.
C.
D.
E.
F.
A
f(z)= C
if B
if D
E if F
G. What is the cost if you purchased 40 shirts?
Answer:
Step-by-step explanation:
i don’t know
Answer:i really don´t know hopefully you get thw\e answer correct.
Step-by-step explanation:
Audrey’s university English teacher Mr. Dalton has told her she will have 28 assignments this semester. Audrey has discovered that she needs to spend 2.5 hours on each assignment.
How many hours will Audrey spend on her English assignments?
AND
How many days will it take Audrey to complete her assignments?
I need explanation please!
70 hours will Audrey spend on her English assignments.
2.9 days will it take Audrey to complete her assignments.
What is Unitary Method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.
For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.
12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.
As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.
This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.
Given:
Mr. Dalton has told her she will have 28 assignments this semester.
She needs to spend 2.5 hours on each assignment.
So, she will spend on Assignments
= 28 x 2.5
= 70 hours
Now, In one day = 24 hours
So, to complete the assignments it will take
= 70/ 24
= 2.9 days
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Someone help me please, I am struggling with this
Answer:
79° , 65° , 144°
Step-by-step explanation:
65° , x° and 36° lie on a straight line and sum to 180° , that is
65° + x° + 36° = 180°
101° + x° = 180 ( subtract 101° from both sides )
x= 79°
y + 10 and 65° are alternate angles and are congruent , then
y + 10 = 65°
z and 36° are same- side interior angles and sum to 180° , that is
z + 36° = 180° ( subtract 36° from both sides )
z = 144°
Identify which graph can be used to solve each equation. Enter the letter of the correct graph next to the
equation.
A
DONE
SL
30
20
10
21
x + 3 = 0
B
St
20
10
10
4
2
(x-3)4 = 0
C
S
30
20
10
2
(x²-3)² = 0
The lengths of RS and ST are 20 and 1 respectively
What is length?Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units system the base unit for length is the metre.
here, we have,
to solve for RS and ST;
The given parameters are:
RS= 2x+10, ST= x−4, RT= 21
This means that
RT = RS + ST
So, we have:
2x + 10 + x - 4 = 21
Evaluate the like terms
3x = 15
Divide by 3
x = 5
Substitute x = 5 in RS= 2x+10 and ST= x−4
RS= 2*5+10 = 20
ST= 5−4 = 1
Hence, the lengths of RS and ST are 20 and 1 respectively
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Find the exact value of each expression. (a) cos 165° (b) cos 80° cos 20° + sin 80° sin 20°
(a) We can use the identity cos(180° - θ) = -cos(θ) to rewrite cos(165°) as cos(180° - 15°) cos(165°) = cos(180° - 15°) = -cos(15°) To find cos(15°),
we can use the half-angle formula for cosine: cos(15°) = cos(30°/2) = sqrt((1 + cos(30°))/2) = sqrt((1 + sqrt(3)/2)/2) = (sqrt(2) + sqrt(6))/4
Therefore, cos(165°) = -cos(15°) = -(sqrt(2) + sqrt(6))/4 (b) Using the product-to-sum identity, we have cos(80°)cos(20°) + sin(80°)sin(20°) = cos(80° - 20°) = cos(60°) = 1/2 Therefore, cos(80°)cos(20°) + sin(80°)sin(20°) = 1/2.
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A "pool toy" is thrown into a swimming pool, but floats on the surface of the water. It displaces 200mL of water in the pool. Use this information to select a correct conclusion.
(a) The toy weighs 200 grams. (b) The toy absorbed 200 mL of water. (c) The toy has a surface area of 200 cm². (d) The toy has a volume of 200 cm³.
The toy has a volume of 200 cm³. The correct answer is Option d.
This is because when an object is submerged in water, it displaces an amount of water equal to its volume. In this case, the pool toy displaces 200 mL of water, which means it has a volume of 200 cm³ (since 1 mL is equal to 1 cm³). The correct answer is Option d.
It is not correct to say that the toy weighs 200 grams (option a), as the weight of the toy is not related to the amount of water it displaces. Similarly, it is not correct to say that the toy absorbed 200 mL of water (option b), as the toy is simply displacing the water, not absorbing it.
Finally, it is correct to say that the toy has a surface area of 200 cm² (option c), as the amount of water displaced is related to the toy's volume, not its surface area.
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Find the median of the solutions to the equation 3x^3+5x^2−6x−10=0. (A) −√2 (B) √2 (C) −5/9 (D) −3/5 (E) None of these
The median of the solutions to the equation 3x³ + 5x² − 6x − 10 = 0 is -√2. The correct answer is A.
To find the median of the solutions to the equation 3x³ + 5x² − 6x − 10 = 0, first find the solutions of the equation and then find the middle value of those solutions.
Group the terms and factor out (3x + 5).
(3x³ + 5x²) + (-6x − 10) = 0
(3x + 5)(x²) + (3x + 5)(-2) = 0
(3x + 5)(x² - 2) = 0
Factor (x² - 2).
(3x + 5)(x + √2)(x - √2) = 0
So the solutions of the equation are x = -5/3, x = -√2, and x = √2.
To find the median of these solutions, we need to order them from least to greatest:
-5/3 < -√2 < √2
The median of these solutions is -√2.
Therefore, the correct answer is (A) -√2.
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How many yards of fabric is needed if a choir robe needs 1 2/9 yards of fabric and John plans on making 24 choir robes
Therefore , the solution of the given problem of unitary method comes out to be 29 1/3 yards of fabric.
Describe the unitary method.To finish the job using the unitary method, multiply the measures taken from this microsecond section by two variable. In a nutshell, when a wanted thing is present, the characterized by a group but also colour groups are both eliminated from the unit technique. For expression instance, 40 changeable-price pencils would cost Inr ($1.01). It's conceivable that one nation will have complete control over the strategy used to achieve this. Almost all living things have a unique trait.
Here,
If 1 2/9 yards of cloth are required for each choir robe, then 24 choir robes will require:
=> 1 2/9 * 24 = (1*9+2)/9 * 24 = 11/9 * 24
We add the numerators and denominators together to multiply fractions:
=> 11/9 * 24/1 = (1124)/(91) = 264/9
264/9 yards of cloth will be required to make 24 choir robes. This can be stated simply as:
(Rounded to the closest 1/3 yard) 264/9 = 29 1/3 yards of fabric.
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Let X be a random variable whose characteristic function ∅ satisfies ∫▒〖|∅(t)|dt< [infinity].〗 . Show that (2π)^(-1) ∫▒〖e^(-√(-1) xt)∅(t)dt〗 is the Lebesgue density of X.
(2π)^(-1)E[∫▒〖e^(-√(-1) xy)]dy〗
The characteristic function of a random variable X is defined as ∅(t) = E[e^(√(-1)tx)]. To show that (2π)^(-1) ∫▒〖e^(-√(-1) xt)∅(t)dt〗 is the Lebesgue density of X, we need to demonstrate that it is a probability density function.
We first calculate the integral to obtain (2π)^(-1) ∫▒〖e^(-√(-1) xt)∅(t)dt〗 = (2π)^(-1) ∫▒〖E[e^(-√(-1) xt +√(-1) tx)]dt〗.
Since the expectation is a constant, we can pull it out of the integral to get (2π)^(-1)E[∫▒〖e^(-√(-1) xt +√(-1) tx)]dt〗.
Now, if we substitute t = y-x and rewrite the integral, we obtain (2π)^(-1)E[∫▒〖e^(-√(-1) x(y-x))e^(√(-1) xy)]dy〗.
This simplifies to (2π)^(-1)E[∫▒〖e^(-√(-1) xy)]dy〗.
Because the integrand is 1, the integral is simply y. Then, the expectation becomes E[y] which is the mean of the random variable X. Thus, the Lebesgue density of X is (2π)^(-1)E[y], which is the probability density function of X.
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if
a real estate office sells 1.6 houses on an average weekday and
sales of houses on weekdays are poison distributed. What is the
probability of selling exactly 4 hours in one day ?
The probability of selling exactly 4 houses in one day is 0.0548, or about 5.48%.
To find the probability of selling exactly 4 houses in one day, use the Poisson distribution formula:
P(X = x) = (λ^x)(e^-λ) / x!
Where:
λ = the average number of events (in this case, the average number of houses sold)
x = the number of events we want to find the probability of (in this case, 4 houses)
e = the base of the natural logarithm (approximately 2.71828)
So, plugging in the values: P(X = 4) = (1.6^4)(e^-1.6) / 4! = (6.5536)(0.2019) / 24 = 0.0548
Therefore, since the data is Poisson distributed, selling exactly 4 houses in one day has a probability of 0.0548, or about 5.48%.
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There are 100 prize tickets in a bowl, numbered 1-100. What is the probability that an even numbered prize will be chosen at random, not replaced, then an odd numbered prize ticket will be chosen?
Answer:
25/99
Step-by-step explanation:
If the tickets are numbered 1-100, half of the tickets will be even and half of the tickets will be odd
Number of even tickets = 50
Number of odd tickets = 50
Let A be the event => even ticket on first draw
Let B be the event => odd ticket on second draw
P(even on first draw) = P(A)
= Number of even tickets/total number of tickets
= 50/100
= 1/2
Once a ticket has been drawn and the second draw is without replacement,
Total number of tickets remaining = 100 - 1 = 99
Total number of odd tickets remaining given first ticket drawn is even
= 50
P(odd ticket second draw | first draw is even) = P(B | A)
= 50/99
P(A and B) = P(A) · P(B | A)
= 1/2 x 50/99
= 50 / (2 x 99)
= 50/198
= 25/99
Probability that it is an even number:
Total outcome = 100Favourable outcome = 50 (because there are 50 even numvers and 50 odd numbers)[tex] \tt \: P(E) = \frac{F.O.}{T.O.} [/tex]
[tex] \tt \: P(E) = \frac{50}{100} = \frac{1}{2} [/tex]
Pribability that it is an odd number:
Total outcome = 99 (because one ticket was taken out and was not replaced)Favourable outcome = 50[tex] \tt \: P(E) = \frac{F.O.}{T.O.} [/tex]
[tex] \tt \: P(E) = \frac{50}{99} = 0.505051[/tex]
The regular price of a pair of sneakers is 40 dollars. The sale price is 25 percent off. what is the sale price?
Answer:
30
Step-by-step explanation:
40/4=10
25%=10
40-10=30
0.4 in sin degrees in trigonometry
The value of angle XYZ is 0.020 degrees using the cosine function.
What are inverse trigonometric functions?Simply put, inverse trigonometric functions are the opposites of the fundamental trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant. The terms arcus functions, antitrigonometric functions, and cyclometric functions are also used to describe them. To find the angle for any trigonometric ratio, apply these inverse trigonometric functions. The fields of engineering, physics, geometry, and navigation all heavily utilise the inverse trigonometry functions.
From the given right triangle we observe that:
cos y = 6 / 15 = adjacent side /hypotenuse
y = arccos(6/15)
y = 0.020 degrees
Hence, the value of angle XYZ is 0.020 degrees using the cosine function.
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Suppose a city with population 500,000 has been growing at a rate of 2% per year. If this rate continues, find the population of this city in 25 years
[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &500000\\ r=rate\to 2\%\to \frac{2}{100}\dotfill &0.02\\ t=\textit{elapsed time}\dotfill &25\\ \end{cases} \\\\\\ A = 500000(1 + 0.02)^{25} \implies A=500000(1.02)^{25}\implies A \approx 820303[/tex]
Find a function f whose graph is a parabola with the given vertex and that passes through the given point. vertex (−1, 3); point (−2, −2)
Answer:
Equation of parabola in vertex form:
y = -5( x + 1)² + 3
Equation of parabola in standard form:
y = -5x² -10x - 2
Step-by-step explanation:
The vertex form equation for a parabola is
y = a(x - h)² + k
where (h, k) is the vertex and a is a constant
Given (h, k) = (- 1, 3)
h = -1 , k = 3
y = a( x - (-1) )² + 3
y = a(x + 1)² + 3
To find a, we know the parabola passes through point(- 2, - 2). Plug in x = -2, y = -2 and solve for a
- 2 = a( -2 + 1)² + 3
-2 = a(-1)² + 3
-2 = a · 1 + 3
-2 = a + 3
Subtract 3 from both sides:
-2 - 3 = a + 3 -3
-5 = a
or
a = -5
Equation of parabola in vertex form:
y = -5( x + 1)² + 3
The standard form of this parabola is y = ax² + bx + c
Expand (x + 1)² = x² + 2 · 1 · x + 1² = x² + 2x + 1
Therefore
y = -5( x + 1)² + 3 becomes
y = -5( x² + 2x + 1) + 3
y = -5x² -10x - 5 + 3
y = -5x² -10x - 2
FAMILY The table below shows the predicted annual cost for middle income family to raise a child from birth until adulthood. Draw a scatter plot and describe what relationship exists within the data
you have too draw a scatter
Solve the following system of linear equations by the Gauss-Jordan elimination method 7x7 + 7x4 + 7x3 = 0 2x: + 2x2 - 4x + 6x4 - 2xy = 0 X1 + X2 - 2x3 - x5= 0 + x = 0
2X1 + 2X2 - x3 + x5= 0
The Gauss-Jordan elimination method can be used to solve this system of linear equations. First, we need to create an augmented matrix containing all the coefficients of the system:
\begin{bmatrix}
7 & 7 & 7 & 0 \\
2 & 2 & -4 & 6 \\
1 & 1 & -2 & -1 \\
0 & 0 & 1 & 0 \\
\end{bmatrix}
We then use a series of row operations to reduce the matrix to row echelon form:
\begin{bmatrix}
7 & 7 & 7 & 0 \\
0 & -4 & -5 & 6 \\
0 & 1 & -3 & -1 \\
0 & 0 & 1 & 0 \\
\end{bmatrix}
Finally, we solve the system of equations by performing back substitution and we get the following solution:
x1 = 0, x2 = 6, x3 = 1, x4 = -2, x5 = 1
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Please help quickly! Both circles have the same center. The circumference of the inner circle is 125.6 inches. What is the area of the shaded region?
Use 3.14 for pi. Write your answer as a whole number or decimal rounded to the nearest hundredth.
Answer: 43.96 inches^2
Step-by-step explanation:
After a number of months, a bank balance was -$46. 83. Each month, the bank charges a $5. 95 fee for having a negative balance. Write an inequality to determine at most how many months the fee was charged. Let x be how many months the fee was charged
If x is the number of months the fee was charged, then the inequality to determine at most months the fee was charged is x ≥ (y + 46.83)/5.95 .
Let x be the number of months the fee was charged.
Let y be the initial balance before the fee was charged.
So, we can write the inequality as : y - 5.95x ≤ -46.83
where y is = positive number representing the initial balance.
Since the initial balance is "y" , we can determine the value of x that satisfies this inequality.
The inequality is represented as : x ≥ (y + 46.83)/5.95 .
Therefore, the inequality to determine number of months is x ≥ (y + 46.83)/5.95.
Learn more about Inequality here
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Four more than the quotient of a number and 8 is equal to 2