The exact value of cosine is 4/5
How to determine the exact value of cosineFrom the question, we have the following parameters that can be used in our computation:
(8,-6) be a point on the terminal side of an angle
Using the above as a guide, we have the following:
Cos(angle) = x/(√x² + y²)
In this case
x = 8 and y = -6
substitute the known values in the above equation, so, we have the following representation
Cos(angle) = 8/(√8² + (-6)²)
So, we have
Cos(angle) = 8/10
Simplify
Cos(angle) = 4/5
Hence, the value is 4/5
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Applying Mathematics in real life context
Informe the coach
Tach
You put two tages in the field. The coordinames for target I (6 Points)
The coordinates for target 2 (4 points). See the figure below.
You asked the players to shoot the ball on the targets 4 times. You added that if
they hit one of the targets, they get the target points
Test 1:
Messi shot the ball. His ball followed the path of the equation where x is the target
points:
1 times
3(3x-4)= 24
6.8 (2.3x + 5.7) = 132.6
Test 2:
Ronaldo shot the ball. His ball followed the path of the equation where y is the
target points:
(6y-18) = 12
8.2 (-5) = -16.4
2 times
2 times
Answer:
Based on the information given, it seems like Messi and Ronaldo were asked to shoot the ball at two targets, target 1 and target 2, and the points they earned would be based on which target they hit.
In Test 1, Messi's first shot followed the equation 3(3x-4) = 24. To determine which target he hit, we need to solve for x. Using algebra, we can simplify the equation to 9x - 12 = 8, and then add 12 to both sides to get 9x = 20. Dividing both sides by 9, we find x = 2.2. Since x represents the number of points earned, this means Messi earned 2.2 points with this shot.
For Messi's second shot, the equation is 6.8 (2.3x + 5.7) = 132.6. To solve for x, we can start by dividing both sides by 6.8 to get 2.3x + 5.7 = 19.5. Subtracting 5.7 from both sides, we find 2.3x = 13.8. Dividing both sides by 2.3, we find x = 6.
In Test 2, Ronaldo's first shot followed the equation (6y-18) = 12. To solve for y, we can add 18 to both sides to get 6y = 30, and then divide both sides by 6 to get y = 5. This means Ronaldo earned 5 points with this shot.
For Ronaldo's second shot, the equation is 8.2 (-5) = -16.4. To solve for y, we need to determine the value of x. Since the equation is just a constant times a negative number, we can simplify it to 8.2 * -5 = -41. This means Ronaldo earned -41 points with this shot.
It's important to note that negative points are not possible in this scenario, as the targets only have positive point values.
Step-by-step explanation:
What is the circumference of this circle?
Use π = 3.14
6 cm
Rafael and Lucy, married taxpayers, each contribute $2,850 to their respective § 401(k) plans offered through their employers. The AGI reported on the couple's joint return is $40,500. Determine their credit for retirement plan contributions (the Saver’s Credit).
Answer:
Rafael and Lucy's credit for retirement plan contributions is $570.
Step-by-step explanation:
The Saver's Credit is a tax credit for low and moderate-income taxpayers who make contributions to a retirement plan. The credit is worth between 10% and 50% of the taxpayer's contributions, up to a maximum credit of $1,000 ($2,000 for married taxpayers filing jointly).
To determine the credit amount, we need to calculate the taxpayers' adjusted gross income (AGI) and their credit rate.
For the tax year in question, the AGI is $40,500 and the contribution limit is $2,850 per person. The total contribution amount is $2,850 x 2 = $5,700.
The credit rate is determined by the taxpayer's AGI and filing status, as follows:
For AGI up to $19,750 for married taxpayers filing jointly, the credit rate is 50%.
For AGI between $19,750 and $32,000 for married taxpayers filing jointly, the credit rate is 20%.
For AGI between $32,000 and $40,000 for married taxpayers filing jointly, the credit rate is 10%.
For AGI above $40,000 for married taxpayers filing jointly, the credit is not available.
Since the taxpayers' AGI is $40,500, the credit rate is 10%. The credit amount is calculated as follows:
Credit amount = Credit rate x Contribution amount
Credit amount = 10% x $5,700 = $570
So, Rafael and Lucy's credit for retirement plan contributions is $570.
Kayla and Kevin have the same-size notebooks.
Kayla covered 5/6 of hers with stickers. Kevin covered 5/8 of his with stickers. Who covered more of the notebook? Explain your answer.
Solving provided question, we can say that The least common multiple (LCM) of 6 and 8 is 24. So, we can rewrite Kayla's fraction as 20/24 and Kevin's fraction as 15/24.
What is least common multiple?The least common multiple, or the least common multiple of the two integers a and b, is the smallest positive integer that is divisible by both a and b. LCM stands for Least Common Multiple. Both of the least common multiples of two integers are the least frequent multiple of the first. A multiple of a number is produced by adding an integer to it. As an illustration, the number 10 is a multiple of 5, as it can be divided by 5, 2, and 5, making it a multiple of 5. The lowest common multiple of these integers is 10, which is the smallest positive integer that can be divided by both 5 and 2.
To compare the proportion of each notebook covered with stickers, we need to find a common denominator for the fractions.
The least common multiple (LCM) of 6 and 8 is 24. So, we can rewrite Kayla's fraction as 20/24 and Kevin's fraction as 15/24.
So, Kayla covered more of her notebook with stickers, as 20/24 is greater than 15/24.
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The Belmont race track known as "BIg Sandy" is 1.5 miles long. In 1973, Secretariat won the Belomont Stakes race in 2 minutes and 30 seconds. Assuming he ran on "Big Sandy", what was his unit speed in miles per hour
Answer:
Below
Step-by-step explanation:
If he ran ONCE around the track he ran 1.5 miles in 2 .5 minutes
We will need to use a conversion factor of 60 min / 1 hr :
1.5 mi / 2.5 min * 60 min / hr = 1.5 *60 / 2.5 m/hr = 36 m/hr
If one factor of 2 numbers is 2860 is 13,what is the other factor
The factors of numbers let us know that the other factor of 2860 is 220.
Factors of a number.Multiples and factors are connected to one another. The number that divides a given quantity exactly, leaving no residue, is said to be a factor of the quantity.
We may write a × b = c to represent any given integer. c is a multiple of "a" and "b". "a" and "b" are factors of c according to the concept of factors and multiples since c may be divided by both a and b.
Given there are two numbers in which one factor is 13 and the multiple is 2860, then we can write it as:
13 × b = 2860
where;
b = the other factor
b = 2860/13
b = 220
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Find the perimeter of the triangle as both a decimal and in simplest
radical form
Add the sides' lengths to find a triangle's perimeter. As an illustration, the perimeter of a triangle with sides a, b, and c is given by P = a + b + c.
What is meant by perimeter?The radius of a shape's edge is known as the perimeter. Discover how to calculate a shape's perimeter by multiplying its side lengths.It is measured from the object's edge to its perimeter. Like the yard that is fenced in at your home. The distance around anything is called its perimeter. The fence is 200 feet long for a yard that is 50 feet by 50 feet.The measurement of a shape's perimeter is its length. It takes adding the lengths of all four sides to determine a rectangle's or square's perimeter.The length of a two-dimensional shape's perimeter is referred to as its perimeter. The total length of all the sides of the thing is also a definition for it.To learn more about perimeter refer to:
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for which values of k are the expressions 12d + 8 and 4(3d + 5) - 8 equivalent
A: No values
B. When d = 0
C: when d = 1
D.All values
Answer: A) No values of k are the expressions 12d + 8 and 4(3d + 5) - 8 equivalent.
What does "equivalent value" mean?They are the same if one amount or value has the same value as another.
We need to set the formulae 12d + 8 and 4(3d + 5) - 8 equal to one another and solve for k to get the values of k for which they are equivalent:
12d + 8 = 4(3d + 5) - 8
When we simplify this equation, we obtain:
12d + 8 = 12d + 12
When we take away 12d from both sides, we obtain:
8 = 12
There are no variables of k for which the expressions are similar because this equation is false for all values of d.
As a result, the response is A) No values.
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Write out all functions ????:{1,2,3,4}→{????,????} (using two-line notation). How many functions are there? How many are surjective? How many are injective? How many are bijective?
Since the domain has 4 elements and the range has 2 elements, each element in the domain can be mapped to one of the two elements in the range. Thus, there are a total of [tex]$2^4 = 16$[/tex] functions from the domain to the range.
What is functions ?
In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the codomain or range) with the property that each input is related to exactly one output. Functions can be represented in many ways, including algebraic expressions, graphs, tables, or verbal descriptions.
Functions are widely used in various fields of mathematics, science, and engineering to model and analyse real-world phenomena, and to make predictions or decisions based on data.
There are 4 elements in the domain and an empty set in the codomain.
Using two-line notation, the functions are:
{1→∅, 2→∅, 3→∅, 4→∅}{1→∅, 2→∅, 3→∅, 4→∅}{1→∅, 2→∅, 3→∅, 4→∅}{1→∅, 2→∅, 3→∅, 4→∅}There is only one possible image for each element in the domain, which means that there is only one function.
Since there is only one function, it is both surjective and injective. Therefore, there is only one bijective function.
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An office block has five floors (ground, 1, 2, 3 and 4), all connected by a lift. When it goes up
to any floor (except 4), the probability that after it has stopped it will continue to rise is 3/4.
When it goes down to any floor (except the ground floor), the probability that after it has
stopped it will continue to go down is 1/4. The lift stops at any floor it passes.
The lift is currently at the first floor having just descended. Calculate the probability of
these events.
a Its second stop is the third floor.
b Its third stop is the fourth floor.
c Its fourth stop is the first floor.
Answer:
Step-by-step explanation:
a) The second stop is the third floor:
The first stop is the first floor, so the probability of going up from the first floor is 3/4.
The next stop is the third floor, so the probability of going up from the second floor (which is the first floor in this scenario) to the third floor is 3/4.
The probability of both events happening is the product of the individual probabilities: (3/4) * (3/4) = 9/16.
b) The third stop is the fourth floor:
The first stop is the first floor, so the probability of going up from the first floor is 3/4.
The next stop is the third floor, so the probability of going up from the first floor to the third floor is 3/4.
The next stop is the fourth floor, so the probability of going down from the third floor to the fourth floor is 1/4.
The probability of all three events happening is the product of the individual probabilities: (3/4) * (3/4) * (1/4) = 3/16.
c) The fourth stop is the first floor:
The first stop is the first floor, so the probability of going up from the first floor is 3/4.
The next stop is the third floor, so the probability of going up from the first floor to the third floor is 3/4.
The next stop is the fourth floor, so the probability of going down from the third floor to the fourth floor is 1/4.
The next stop is the first floor, so the probability of going down from the fourth floor to the first floor is 3/4.
The probability of all four events happening is the product of the individual probabilities: (3/4) * (3/4) * (1/4) * (3/4) = 9/64.
A student moves into a bigger apartment that rents for $750 per month. That rent is $200 less than twice what she had been paying. Find her former
rent in dollars.
Answer:
$475
Step-by-step explanation:
x = former rent
2x - 200 = 750
2x = 750 + 200 = 950
x = 950/2 = 475
what is the inequality shown
The inequality on the graph is written as:
-4 < x ≤ 5
What is the inequality shown on the number line?
Here we have an inequality graphed on a number line.
First, we can see that we have an open circle at x = -4 and then the line goes to the right.
That means that the value of x must be larger than -4, so we can start by writting:
-4 < x
Then we have a closed circle at x = 5.
That means that x can be smaller or equal to 5, then:
x ≤ 5
Then the inequality is:
-4 < x ≤ 5
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Write the equation is slope intercept form given
f(0) = -3 and f(1) = 2
The equation is slope-intercept form is equal to y = 5x - 3.
What is the point-slope form?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
Where:
m represents the slope.x and y are the points.Based on the information provided, we can logically deduce the following data points on the line:
Points on x-axis = (0, 1).Points on y-axis = (-3, 2).At point (0, -3), a linear equation of this line can be calculated in slope-intercept form as follows:
y - (-3) = (2 + 3)/(1 - 0)(x - 0)
y + 3 = 5x
y = 5x - 3.
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what formula is used to determine the percentage change in quantity demanded
Answer:
slope
Step-by-step explanation:
Step-by-step explanation:
always find the quantity of 100%.
then
1% = 100%/100
then calculate the absolute quantity change.
then divide the quantity change by 1% to see how many % fit into it.
and that is your answer.
so, if the original quantity is q1, the new quantity is q2, then the % of quantity change is
|q1 - q2| / (q1/100)
or
100 × |q1 - q2| / q1
where || means the absolute value.
Which of the following are equivalent to x2 – 4x + 9? Select four answers.
A. (x – 3)2 – 4x
B. (3x2 – 3x + 1) – (2x2 + x – 8)
C. (x – 3)(x + 3) – 2x
D. (x – 3)2 +2x
E. (x – 5)(x + 1) +14
F. (8x2 + 2x) – (4x2 + 6x) + (9 – 3x2)
Answer:
a).2x-6-5x-12x. this is answer
The hypotenuse of a right triangle has a length of 14 units and a side that is 9 units long. Which equation can be
used to find the length of the remaining side?
The equation that can be used to find the length of the remaining side is z² = 14² - 9²
How to determine the length of sideUsing the Pythagorean theorem that states that the square of the hypotenuse side is equal or equivalent to the sum of square of the other two sides.
This is represented as;
x² = y² + z²
Given that the parameters are enumerated as;
x is the hypotenuse sidey is the opposite sidez is the adjacent sideNow, substitute the values
14² = 9² + z²
collect like terms
z² = 14² - 9²
Hence, the equation is z² = 14² - 9²
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The length of the remaining side can be calculated using 14² = x² + 9² is 10.7 units
What is an equation?An equation is an expression that shows how numbers and variables are linked together using mathematical operations such as addition, subtraction, multiplication and division.
Pythagoras theorem is used to show the relationship between the sides of a right angled triangle. It is given as:
hypotenuse² = adjacent² + opposite²
Let x represent the missing side.
The hypotenuse of a right triangle has a length of 14 units and a side that is 9 units long. Hence:
14² = x² + 9²
x² = 14² - 9²
x² = 115
x = 10.7 units
The length of the remaining side is 10.7 units
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What is the result when the number 45 is increased by 57%?
Answer: The result when the number 45 is increased by 57% is approximately 70.65.
Step-by-step explanation:
To find the result when the number 45 is increased by 57%, we need to calculate 45 * (1 + 0.57), where 0.57 represents 57% as a decimal.
Performing the calculation, we get:
45 * (1 + 0.57) = 45 * 1.57 = 70.65
So, the result when the number 45 is increased by 57% is approximately 70.65.
Answer:
70.65
Step-by-step explanation:
The new value =
45 + The value of the percentage increase =
45 + (57% × 45) =
45 + 57% × 45 =
(1 + 57%) × 45 =
(100% + 57%) × 45 =
157% × 45 =
157 ÷ 100 × 45 =
157 × 45 ÷ 100 =
7,065 ÷ 100 =
70.65
Jean bought a $1,980 snow thrower on an installment plan. The installment agreement included a 10% down payment and 18 monthly payments of $116 each.
Jean purchased a snow thrower for $1,980 using an installment plan that required a 10% down payment of $198 and 18 monthly payments of $116. This allowed Jean to spread out the cost of the snow thrower over time, avoiding a large upfront cost.
Jean recently purchased a snow thrower for $1,980 using an installment plan. To start, Jean had to make a down payment of 10% of the total purchase price, which was $198. This down payment allowed Jean to spread the remaining cost of the snow thrower across 18 monthly payments of $116 each. After making all 18 payments, Jean will have paid the full purchase price of $1,980. This installment plan option is an excellent way for Jean to get the snow thrower he needs while avoiding a large upfront cost and spreading the payments out over a period of time. With this installment plan, Jean can pay off the purchase of the snow thrower while still having money left over each month to cover other expenses.
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Factorise: 16x4 - y4
Answer:
= (4x^2 + y^2)(2x + y)(2x - y).
Step-by-step explanation:
16x^4 - y^4
= (4x2)^2 - (y^2)^2
= (4x^2 +y^2)(4x^2 - y^2)
=(4x^2 + y^2)( (2x)^2 -y^2)
= (4x^2 + y^2)(2x + y)(2x - y).
Answer:
Step-by-step explanation: [tex]16x^{4} -4y^{4} = (4x^{2} )^{2} -(2y^{2} )^{2}= (4x^{2} + 2y^{2} )(4x^{2} - 2y^{2})[/tex]
identity used =
[tex](a^{2} -b^{2} ) = (a+b)(a-b)[/tex]
I need the answer NO LINKS URGENT NO LINKS URGENTS
Answer: See the attached image below for the proof table.
Explanation:
We start with what we're given. Simply repeat the given information word for word. This is statement 1 of the proof.
Statement 2 mentions AB is parallel to CD. This is valid due to the definition of a parallelogram. The name means "two sets of opposite parallel sides. Statement 3 is a similar situation.
Statement 4 relies on statement 2. The red angles in the diagram are congruent alternate interior angles. They are only congruent because of AB parallel to CD.
Statement 5 relies on statement 3 for similar reasoning as the previous paragraph. Focus on the blue angles.
Statement 6 is hopefully self explanatory so there's not much for me to mention here.
Statement 7 has us connect statements 4, 5, and 6 to prove triangle ADB is congruent to triangle CBD. We use the ASA (angle side angle) theorem here.
Statements 8 and 9 finish up the proof. CPCTC stands for "Corresponding parts of congruent triangles are congruent". If two triangles are congruent, then their corresponding pieces must be congruent as well. It would be like saying two houses are identical if and only if their front doors are identical.
5.
Jenny had to increase her gas budget from $300 per month to $350 per
month. What is the percentage increase in Jenny's budget?
Answer:
16.66667% increase
Step-by-step explanation:
100 x 350 - 300/ 300 = 50/3 = 16.66667% increase
Formula for percent increase is below↓
Give the most specific name for the quadrilateral.
O parallelogram
rectangle
O trapezoid
O isosceles trapezoid
Explain your reasoning.
B I U
E T² T₂
The most specific name for the quadrilateral is trapezoid, the correct option is C.
What are quadrilateral?A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four angles. There are two main types: concave and convex. There are also various subcategories of convex quadrilaterals, such as trapezoids, parallelograms, rectangles, rhombi, and squares.
Given;
The figure with 4 sides and two opposite are equal
Now,
In the figure two opposite sides are equal in length
But they all are not parallel only on pair of opposite sides are parallel.
Also, the figure the adjacent angle are equal.
Therefore, the quadrilateral will be trapezoid.
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Consider the expressions {4x^3+2x^2+6x+7}/{2x+1} and 2x^2+3+(4/{2x+1}.)
a) Show that these expressions are equal when x=10.
b) Explain why these expressions are not equal when x=-1/2.
c) Show that these expressions are equal for all x other than-1/2.
In parts (a) and (c), begin by explaining what your strategy for solving will be.
The given expressions {4x^3+2x^2+6x+7}/{2x+1} and 2x^2+3+(4/{2x+1}) are equal at x = 10, they are not equal at x = -1/2, and they are equal for all x other than -1/2.
Substitute x = 10 in the given expressions, we get
{4x^3+2x^2+6x+7}/{2x+1}
= (4000+200+60+7)/21
=203.19
2x^2+3+(4/{2x+1}.)
200+3+4/21
=203.19
Hence, the given expressions are equal when x = 10
Now substitute x = -1/2 in the given expressions
={4x^3+2x^2+6x+7}/{2x+1}
=(4*-1/8+2*1/4+6*-1/2+7)/0
2x^2+3+(4/{2x+1}.)
=2*1/4+3+4/0
When we substitute the value x= -1/2 then it divide by 0 and value will be infinite and hence we can say that the values are not equal at x = -1/2
2x^2+3+(4/{2x+1}.) take LCM
{4x^3+2x^2+6x+7}/{2x+1} here both expression are equal and we can't able find the value at x = -1/2 but for other value of x they are equal.
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12 (long division) 924
The solution is, 12 (long division) 924 = 77.
What is division?Division is the process of splitting a number or an amount into equal parts.
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication.
here, we have,
given that,
12 (long division) 924
so, we get,
12 | 924 | 77
84
----------------
84
84
-----------------
0
Hence, The solution is, 12 (long division) 924 = 77.
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For four months, you have saved $25 per month.
You now have $175 in your savings account.
a. Use your result from Exploration 2 to
write an equation that represents the
balance A after t months.
Answer:
A=25t.
Step-by-step explanation:
..........................
Leyan bought 2 3/4 pounds of apples for $1.32 per pound, 1 1/2 pounds of peaches for $1.20 per pound, and some tomatoes, all from a grocery store. The total cost of his purchase was $9.63.
How much money did Leyan spend on tomatoes?
The diagonals of parallelogram ABCD intersect at P. Which statements must be true? Select all that apply.
The statement that must be true for the parallelogram is
D. < CAD is congruent to < ACBWhat is alternate internal angles?Alternate internal angles are pairs of angles that lie on opposite sides of a transversal line intersecting two parallel lines. The angles are on the interior (between the two parallel lines) and are not adjacent to each other.
These angles are equal in measure (i.e., they are congruent), which means that if one angle has a certain degree of measure, the other angle will have the same degree of measure.
In the problem, the diagonal of the parallelogram is the transversal line and it is used to make the alternate internal angle theorem possible
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You can earn a maximum profit of $
by investing $
in stock A, $
in stock B, and $
in stock C.
The solution that gives the maximum profit for the linear optimization question is presented as follows;
A maximum profit of $840 can be earned by investing;
$3,000 in stock A
$6,000 in stock B
$7,000 in stock C
What is linear programming?Linear programming is a optimization technique that helps in determining the maximizing or minimizing parameters of linear functions.
The parameters in the question, obtained from a similar question obtained online includes;
The amount available to be invested = $16,000
The expected returns for stock A = 4%
The expected return for stock B = 5%
Expected returns for stock C = 6%
Amount to be invested in each stock ≥ $1,000
Amount to be invested in stock C ≤ $7,000
Amount invested in C ≤ 4 × Amount invested in stock A
Amount invested in stock B ≤ 2 × Amount invested in stock A
Let x represent the amount invested in stock A, let y represent the amount invested in stock B therefore;
The amount invested in stock C = 16,000 - x - y
x ≥ 1000y ≥ 1000y ≤ 2·x16,000 - x - y ≥ 1000...(1)
16,000 - x - y ≤ 7000...(2)
16,000 - x - y ≤ 4·x...(3)
Therefore, we get;
- y ≥ 1000 - 16,000 = -15000 + x
-y ≥ -15000 + x
y ≤ 15000 - x16,000 - x - y ≤ 7000
y ≥ 9,000 - x16,000 - x - y ≤ 4·x
-y ≤ 4·x + x = 5·x - 16,000
-y ≤ 5·x - 16,000
y ≥ 16,000 - 5·xThe vertices of the feasible regions on the graph of the inequalities are;
(5,000, 10,000), (3,000, 6,000), (8,000, 1,000), and (14,000, 1,000)
The profit function is;
Profit = 0.04·x + 0.05·y + 0.06·(16000 - x - y)
Plugging in the value of the vertices of the feasible region in the profit function, we get;
Point (5,000, 10,000)
Profit = 0.04 × 5000 + 0.05 × 10000 + 0.06 × (16000 - 5000 - 10000) = 760
Point (3,000, 6,000)
Profit = 0.04 × 3000 + 0.05 × 6000 + 0.06 × (16000 - 3000 - 6000) = 840
Point (8,000, 1,000)
Profit = 0.04 × 8000 + 0.05 × 1000 + 0.06 × (16000 - 8000 - 1000) = 790
(14,000, 1,000)
Profit = 0.04 × 14000 + 0.05 × 1000 + 0.06 × (16000 - 14000 - 1000) = 670
Therefore, the maximum profit of $840 is earned by investing $3,000 in stock A and $6,000 in stock B which indicates that the investment in stock C is $16,000 - $3,000 - $6,000 = $7,000
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The function f(x) = x³ +9 is one-to-one.
a. Find an equation for f1, the inverse function.
b. Verify that your equation is correct by showing that f(f(x)) = x and f¹(f(x)) = x.
a. Select the correct choice below and fill in the answer box(es) to complete your choice.
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
OA. f¹(x)=, for all x
O B. f¹(x)=, for x ≥
O c. f¹(x)=, for x‡|
x#
OD. f¹(x)=, for xs
Since f(f1(y)) = y and f1(f(x)) = x, we can conclude that f1 is the correct inverse function for f.
What is functio ?
Function can be defined in which it relates an input to output.
Given function ,
The function f(x) = x³ +9 is one-to-one.
To find the inverse function f1, we need to switch the roles of x and y in the equation for f(x). So, we have:
y = x³ + 9
Next, we need to solve for x in terms of y:
x³ + 9 = y
x³ = y - 9
x = (y - 9[tex])^{(1/3)}[/tex]
So, the inverse function is:
f1(y) = (y - 9[tex])^{(1/3)}[/tex]
To verify that f1 is the correct inverse function, we need to show that f(f1(y)) = y and f1(f(x)) = x:
f(f1(y)) = f((y - 9[tex])^{(1/3)}[/tex] = (y - 9[tex])^{(1/3)}^3[/tex] + 9 = y - 9 + 9 = y
And:
f1(f(x)) = f1(x³ + 9) = (x³ + 9 - 9[tex])^{(1/3)}[/tex] = [tex]\mathrm{ x^3^{(1/3)}}[/tex] = x
Therefore, Since f(f1(y)) = y and f1(f(x)) = x, we can conclude that f1 is the correct inverse function for f.
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It is estimated that approximately one-half of all aluminum cans will be recycled each year. If a soft drink company produces 350,000 cans one year, how many cans are still in use after 4 years of recycling and re-using, using this function?
N, = 350,000(0.52)
A.87,500
B.25,591
C.175,000
D.21,875
25,591 cans are still in use after 4 years of recycling and re-using. The solution has been obtained by using functions.
What is a function?
A function is a connection between a set of inputs and one output for each input. The usual notation for a function is f(x), where x represents the input. Usually, a function is written as y = f(x).
We are given a function as N = 350,000 [tex](0.52)^{t}[/tex]
t is given to be 4 years.
Substituting this in the function, we get
⇒N = 350,000 [tex](0.52)^{4}[/tex]
⇒N = 350,000 * 0.07311616
⇒N = 25,590.6
⇒N = 25,591
Hence, 25,591 cans are still in use after 4 years of recycling and re-using.
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