The partial derivatives of f with respect to u and v at (u,v) = (-2, -2) are ∂f/∂u(-2,-2) = 224 and ∂f/∂v(-2,-2) = -1056.
To evaluate the partial derivatives ∂f/∂u and ∂f/∂v at (u,v) = (-2, -2) using the Chain Rule, we need to find the derivatives of f with respect to x, y, and z, and then multiply them by the derivatives of x, y, and z with respect to u and v.
First, let's find the derivatives of f with respect to x, y, and z:
∂f/∂x = 3x^2
∂f/∂y = z^2
∂f/∂z = 2yz
Next, let's find the derivatives of x, y, and z with respect to u and v:
∂x/∂u = 2u
∂x/∂v = 1
∂y/∂u = 1
∂y/∂v = 2v
∂z/∂u = 4v
∂z/∂v = 4u
Now, we can use the Chain Rule to find the partial derivatives of f with respect to u and v:
∂f/∂u = (3x^2)(2u) + (z^2)(1) + (2yz)(4v)
∂f/∂v = (3x^2)(1) + (z^2)(2v) + (2yz)(4u)
Finally, we can plug in the values of u and v to find the partial derivatives at (u,v) = (-2, -2):
∂f/∂u(-2,-2) = (3((-2)^2 + (-2))^2)(2(-2)) + ((4(-2)(-2))^2)(1) + (2((-2)+(-2)^2)(4(-2)))
= (3(0)^2)(-4) + (16^2)(1) + (2(2)(-8))
= 0 + 256 - 32
= 224
∂f/∂v(-2,-2) = (3((-2)^2 + (-2))^2)(1) + ((4(-2)(-2))^2)(2(-2)) + (2((-2)+(-2)^2)(4(-2)))
= (3(0)^2)(1) + (16^2)(-4) + (2(2)(-8))
= 0 - 1024 - 32
= -1056
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You bought water bottles to sell and spent a total of $50. You sell each bottle for $2. Which graph shows how many bottles you have to sell to earn back the money spent on water?
The graph that shows the number of bottles that need to be sold to earn back the money spent on water is Graph C.
How to find the graph ?The number of bottles that need to be sold to earn back the amount spent on water is :
= Total spent / profit per bottle
= 50 / 2
= 25 bottles
Graph C starts from - $ 50 to show that the amount you have is - $ 50 to show the amount spent on the bottles.
Graph C then ends at 25 bottles to show that this is where you would need to stop to have made back what you spent.
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A parking garage in Quincy, IL charges $6.00 for the first hour and $2.00 for each additional hour or part of an hour. What is the maximum number of hours that you can park in the garage if you wish to pay no more than $14?
The maximum number of hours that you can park in the garage if you wish to pay no more than $14 is 5 hours.
Here's how to calculate it:
Start with the first hour charge of $6.00.Subtract this from the maximum amount you want to pay ($14) to get $8 remaining.Divide this remaining amount by the cost of each additional hour ($2) to get the number of additional hours you can park. $8/$2 = 4 additional hours.Add the first hour to the additional hours to get the maximum number of hours you can park. 1 + 4 = 5 hours.Therefore, the maximum number of hours you can park in the garage if you wish to pay no more than $14 is 5 hours.
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Math question 12 help
Answer:
It's neither of them. The function is neither odd nor even
{ 4.82/ 7 } 358.9 / 7 Quotient
Answer:
51.2
Step-by-step explanation:
By dividing :0
Prove that the medial triangle of a given triangle ABC
has sides parallel to and half the length of the sides of triangle
ABC.
The medial triangle of a given triangle ABC has sides parallel to and half the length of the sides of triangle ABC.
To prove that the medial triangle of a given triangle ABC has sides parallel to and half the length of the sides of triangle ABC, we can use the following steps:
Draw triangle ABC and label its vertices as A, B, and C.
Find the midpoints of the sides of triangle ABC. Let's call these midpoints D, E, and F.
Draw the medial triangle DEF, connecting the midpoints of the sides of triangle ABC.
We need to prove that the sides of triangle DEF are parallel to the sides of triangle ABC. To do this, we can use the fact that the midpoints of the sides of a triangle divide the sides into two segments of equal length.
Since the midpoints of the sides of triangle ABC are D, E, and F, we know that AD = DB, BE = EC, and CF = FA.
We can also use the fact that if two lines are parallel and the corresponding angles are equal, then the corresponding sides are also parallel.
Since AD = DB and BE = EC, we know that angle ADB = angle BEC and angle BDE = angle CEF.
Therefore, we can conclude that the sides of triangle DEF are parallel to the sides of triangle ABC.
To prove that the sides of triangle DEF are half the length of the sides of triangle ABC, we can use the fact that the midpoints of the sides of a triangle divide the sides into two segments of equal length.
Since the midpoints of the sides of triangle ABC are D, E, and F, we know that AD = DB, BE = EC, and CF = FA.
Therefore, we can conclude that the sides of triangle DEF are half the length of the sides of triangle ABC.
So, the medial triangle of a given triangle ABC has sides parallel to and half the length of the sides of triangle ABC.
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pls asap i need this
The simplest form of the given expression x will be 42.
What is polynomials?Using variables and coefficients, polynomials are algebraic expressions. The term "indeterminates" is sometimes used to describe variables. The terms Poly and Nominal, which together signify "many" and "terms," make up the word polynomial.
When exponents, constants, and variables are combined using mathematical operations like addition, subtraction, multiplication, and division, the result is a polynomial (No division operation by a variable). The expression is categorized as a monomial, binomial, or trinomial based on the number of terms it contains.
Here we assume that no of suitcases be x
So, 7/24 = x/144
24x = 1008
x = 42.
Hence the simplest form of the given expression x will be 42.
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The height of a pole is 27 feet. A snake is curled up at a distance of 20 ft from the foot of
the pole. The snake looks at the top most point of the pole. Find the angle of elevation
made by the snake and the top of the pole. Round to the nearest tenth of a degree.
The angle of elevation is _____ degrees.
Help
The angle οf elevatiοn is 63.4 degrees.
What is angle οf elevatiοn?Angle οf elevatiοn is the angle between the hοrizοntal line οf sight and an οbject abοve the οbserver. It is measured with an angle starting frοm the hοrizοntal line and increasing in the upward directiοn. It is alsο knοwn as the angle οf elevatiοn οr inclinatiοn. The angle οf elevatiοn is used in navigatiοn, astrοnοmy and trigοnοmetry.
Tο find the angle οf elevatiοn, we can use the fοrmula fοr finding angles in a triangle. This fοrmula states that the sum οf all angles in a triangle is 180 degrees. Tο sοlve fοr the angle οf elevatiοn, we need tο begin by first finding the length οf the οppοsite side οf the triangle. This is dοne by subtracting the distance οf the snake frοm the fοοt οf the pοle frοm the height οf the pοle.
Therefοre, the length οf the οppοsite side οf the triangle is 7 feet. Nοw, we can use the tangent fοrmula tο sοlve fοr the angle οf elevatiοn. This fοrmula states that the tangent οf an angle is equal tο the length οf the οppοsite side divided by the length οf the adjacent side.
Let's find the angle of elevation.
tan ∅ = opposite / adjacent
where
∅ = angle of elevation
Therefore,
tan ∅ = 27 / 20
∅ = tan⁻¹ 1.35
∅ = 53.471144633
∅ = 53.5 degrees
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Mary wins £400.
She gives 50% to her mother.
Mary gives 20% to her sister.
She keeps the rest.
Work out how much each person receives.
Answer:
Mother: £200, Sister: £80, Mary: £120
Step-by-step explanation:
Mary has a total of £400 to start off with.
In order to calculate the amount she gives to her mother, we must multiply 400 by .5 (the decimal equivalent of 50%), which leaves us with £200 to her mother.
Do the same with her sister; multiply 400 by .2 (the decimal equivalent of 20%), which leaves us with £80.
To figure out what she is left with, £400 - 200 - 80 = £120. Mary is left with £120.
Mary will have £120 with her, her mother will have £200 and her sister will have £80.
What is percentage?Percentages are fractions with 100 as the denominator. It is the relation between part and whole where the value of whole is always taken as 100.
Given that, Mary won £400.
She gave 50% to her mother, 20% to her sister and rest she kept with herself,
So,
The amount her mother got =
50% of £400 = 1/2 × 400 = 200
Therefore, her mother got £200.
The amount her sister got =
20% of £400 = 1/5 × 400 = 80
Therefore, her sister got £80.
The amount Mary have = 400 - (200+80) = 400 - 280 = £120
Hence Mary will have £120 with her.
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2
A geometric sequence is shown below.
Which function describes this sequence?
5, 11, 29, 83, ...
These values match the given sequence, so the function that describes the sequence is:
an = 5(2.2 + 0.8182n)ⁿ⁻¹
What is the meaning of the word "function"?According to one definition, a function is a relationship between a number of inputs where each input has exactly one output.
To find the function that describes the given geometric sequence, we need to find the common ratio r. We can do this by dividing any term in the sequence by the previous term:
11/5 = 2.2
29/11 = 2.63636...
83/29 = 2.86206...
We can see that the common ratio is increasing, which suggests that the sequence is not a perfect geometric sequence. However, the differences between the ratios are getting smaller, so we can approximate the sequence as a geometric sequence with a changing common ratio.
Let's use the first two terms of the sequence to find an initial approximation for the common ratio:
r ≈ 11/5 = 2.2
Then, we can write the nth term of the sequence as:
an = 5(2.2)ⁿ⁻¹
However, we noticed that the common ratio is increasing, so we need to adjust our formula to account for this. Let's try a formula where the common ratio is a linear function of n:
an = 5(2.2 + 0.8182n)ⁿ⁻¹
Plugging in n = 1, 2, 3, and 4, we get:
a1 = 5(2.2 + 0.8182(1))⁰ = 5
a2 = 5(2.2 + 0.8182(2))¹ ≈ 11
a3 = 5(2.2 + 0.8182(3))² ≈ 29
a4 = 5(2.2 + 0.8182(4))³ ≈ 83
These values match the given sequence, so the function that describes the sequence is:
an = 5(2.2 + 0.8182n)ⁿ⁻¹
where n is the index of the term in the sequence.
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3. Suzy borrowed R2 400 from a bank for a period of two year and six months at a simple annual interest rate of 4.7% How much must she repay at the end of the period?
The amount Suzy must repay at the end of the period is R2 682.
What is simple interest?Simple interest is a method of calculating the interest charge. Simple interest can be calculated as the product of principal amount, rate and time period.
Simple Interest = (Principal × Rate × Time) / 100
In this case, we have:
P = R2 400
r = 0.047 (since the annual interest rate is 4.7%)
t = 2.5 years (since the period is two years and six months)
Substituting these values into the formula, we get:
I = 2400 * 0.047 * 2.5
I = 282
Therefore, the interest is R282.
The total amount to be repaid is the sum of the principal and the interest, which is:
2400 + 282 = R2 682
Therefore, by the simple interest the answer will be R2 682.
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1) Keith decided to look at new and used SUVs. Keith found a used SUV for $20000. A new SUV is $25000, so what percent of the price of a new SUV does Keith pay for a used SUV? Round your answer to the nearest whole number if necessary.
Keith pays about 80% of the price of a new SUV for the used SUV.
To find what percent of the price of a new SUV Keith pays for a used SUV, we need to divide the price of the used SUV by the price of the new SUV and then multiply by 100 to express the answer as a percentage:
used SUV price / new SUV price * 100%
Substituting the given values, we get:
$20,000 / $25,000 * 100% ≈ 80%
So Keith pays about 80% of the price of a new SUV for the used SUV. Rounded to the nearest whole number, this would be 80%.
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Natalie and johnathan are asked to write an expression equivalent to 2(2x + 2y
The expression equivalent to 2(2x + 2y) are,
⇒ 4 (x + y)
Or, (4x + 4y)
What is an expression?An expression which is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division is called an mathematical expression.
Given that;
The expression is,
⇒ 2 (2x + 2y)
Since, Natalie and Johnathan are asked to write an expression equivalent to 2(2x + 2y).
Hence, We get;
⇒ 2 (2x + 2y)
Take 2 as common;
⇒ 2 × 2 (x + y)
⇒ 4 (x + y)
⇒ 4x + 4y
Thus, The expression equivalent to 2(2x + 2y) are,
⇒ 4 (x + y)
Or, (4x + 4y)
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Eddie says a property of trapezoids is that they have 2 pairs of opposite sides that are parallel. Lacy disagrees. She says Eddie's statement describes an attribute of this trapezoid, not a property of all trapezoids. Who is correct? Explain.
Answer: Lacy is correct. Eddie's statement describes an attribute of a specific trapezoid, not a property of all trapezoids.
A trapezoid is defined as a quadrilateral with at least one pair of parallel sides. Therefore, it is true that all trapezoids have at least one pair of parallel sides. However, not all trapezoids have 2 pairs of opposite sides that are parallel.
The trapezoid that Eddie is describing is a special type of trapezoid called an isosceles trapezoid. An isosceles trapezoid has two pairs of opposite sides that are parallel and congruent legs. However, not all trapezoids are isosceles trapezoids.
Therefore, Lacy is correct in stating that Eddie's statement describes an attribute of a specific trapezoid, not a property of all trapezoids.
Step-by-step explanation:
Determine the coefficient and the degree of each term in the polynomial. 8x^(2)+6x+8
The coefficients of the terms in the polynomial are 8, 6, and 8, and the degrees are 2, 1, and 0
The coefficient of a term in a polynomial is the number that is multiplied by the variable. The degree of a term is the exponent of the variable. In the polynomial 8x^(2)+6x+8, there are three terms:
- The first term is 8x^(2). The coefficient of this term is 8, and the degree is 2.
- The second term is 6x. The coefficient of this term is 6, and the degree is 1.
- The third term is 8. The coefficient of this term is 8, and the degree is 0 (since there is no variable).
Therefore, the coefficients of the terms in the polynomial are 8, 6, and 8, and the degrees are 2, 1, and 0.
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Marco Edwards Question 6 - 3 Points Cob 24, 8:57:21 AM Find the slope of a line perpendicular to the line whose equation is 3x+y=8. oully simplify your answer. Answer: Submit Answer
The slope of a line perpendicular to the line whose equation is 3x+y=8 is 1/3.
The slope of a line perpendicular to the line whose equation is 3x+y=8 can be found by first finding the slope of the given line and then taking the negative reciprocal of that slope.
To find the slope of the given line, we can rearrange the equation to solve for y:
3x+y=8
y=-3x+8
Now we can see that the slope of the given line is -3. To find the slope of a line perpendicular to this one, we take the negative reciprocal of -3, which is 1/3.
Therefore, the slope of a line perpendicular to the line whose equation is 3x+y=8 is 1/3.
Answer: 1/3
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These charts show state sales tax as of 2015. Alabarna Florida Georgia Hawaii Idaho Indiana lowa Kentucky Idaho State Mississippi Hawaii Wisconsin 4% 6% 4% 4% 6% 7% 6% 6% Louisiana Maryland Michigan Mississippi 4% 6% 6% 7% New York North Dakota Oregon Pennsylvania 6% 5% 8 Students from various states are attending a marching band conference. Each uniform costs $135 before state sales tax is added. Complete the table to show the total cost of uniforms with tax, for each group of students. Number of Students 10
5
15 Rhode Island South Carolina South Dakota Vermont West Virginia Wisconsin Wyoming 7% 6% 4% Total Cost of Uniforms
Completing the table to show the total cost of uniforms with tax, for groups of students from the four states is as follows:
State Number Total Cost
of Students (After-tax)
Idaho 10 $1,431.00
Mississippi 5 $722.25
Hawaii 15 $2,106.00
Wisconsin 8 $1,134.00.
How is the total cost for each group determined?The total cost for each group of students from the four states can be determined by multiplication.
The multiplication operation for each group or state first computes the total cost before tax and the total cost after tax.
The total cost after tax uses the sales tax factor or rate to multiply the total cost before tax.
Generally, sales taxes are levied by state and local governments to check the consumption and production of certain items and to increase the revenue of the government.
The price before state sales tax of each uniform = $135
Relevant Taxes and Number of Students:
State Number of Students Sales Tax Rate
Idaho 10 6%
Mississippi 5 7%
Hawaii 15 4%
Wisconsin 8 5%
State Number Total Cost Total Cost
of Students (Before tax) (After-tax)
Idaho 10 $1,350 $1,431.00 ($135 x 10 x 1.06)
Mississippi 5 $675 $722.25 ($135 x 5 x 1.07)
Hawaii 15 $2,025 $2,106.00 ($135 x 15 x 1.04)
Wisconsin 8 $1,080 $1,134.00 ($135 x 8 x 1.05)
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NEEDD HELP ASAP WILL MARK BRAINLIST!! Olivia and Abby are donating part of their allowance to charity. Abby is giving $10
less than twice the amount that Olivia is giving. In total, they are donating $125 to
charity. How much money is Abby giving?
$110
$68
$52
$75
$80
$98
Answer:
80
Step-by-step explanation:
45 x 2 = 90
90 - 10 = 80
80 + 45 = 125
Answer:
$80
Step-by-step explanation:
20% tip, 8% tax sales. Total $23. 04, what is the pretax?
For a 20% tip, 8% tax sales. Total $23. 04, we can say that The pre-tax cost is $19.20.
Let x be the pre-tax cost of the sales. Then we can set up the equation:
x + 0.20x + 0.08x = 23.04
Simplifying and solving for x, we get:
1.28x = 23.04
x = 23.04 / 1.28
x = 18
Therefore, the pre-tax cost of the sales is $18. However, we need to add the 20% tip and 8% tax to this amount to get the total cost of $23.04. The tip is $3.60 (0.20 * 18) and the tax is $1.44 (0.08 * 18), which when added to $18 gives us a total cost of $23.04.
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Question 3 For each of the following, explain whether it is always) true or (possibly) false. If true, you must explain why. If false, you must give a concrete counterexample (with numbers).
a). (1.5pts) If the linear system Ax = b has infinitely many solutions, then 6 is a linear combination of the columns of A. b). (1.5pts) If a 3x3 matrix A has rank 2, then Null(A) is a plane through the origin. c). (1.5pts) If {ū, v} is linearly dependent then so is {ü, v, w}.
a) This system has infinitely many solutions, but 6 is not a linear combination of the columns of A, which are [2, 3] and [4, 6].
b) The null space of the matrix, which is the set of all solutions to the homogeneous equation Ax = 0, is a subspace of dimension 3 - 2 = 1, which is a line through the origin.
c) The set {ū, v} is linearly independent, but the set {ü, v, w} is linearly dependent, as w can be written as a linear combination of ū and v (w = ū + v).
a). (1.5pts) This statement is possibly false. It is not always true that if the linear system Ax = b has infinitely many solutions, then 6 is a linear combination of the columns of A. A concrete counterexample would be the linear system:
2x + 3y = 6
4x + 6y = 12
This system has infinitely many solutions, but 6 is not a linear combination of the columns of A, which are [2, 3] and [4, 6].
b). (1.5pts) This statement is always true. If a 3x3 matrix A has rank 2, then Null(A) is a plane through the origin. This is because the rank of a matrix is the number of linearly independent columns in the matrix, and if the rank is 2, then there are 2 linearly independent columns in the matrix. This means that the null space of the matrix, which is the set of all solutions to the homogeneous equation Ax = 0, is a subspace of dimension 3 - 2 = 1, which is a line through the origin.
c). (1.5pts) This statement is possibly false. It is not always true that if {ū, v} is linearly dependent then so is {ü, v, w}. A concrete counterexample would be the vectors ū = [1, 0], v = [0, 1], and w = [1, 1]. The set {ū, v} is linearly independent, but the set {ü, v, w} is linearly dependent, as w can be written as a linear combination of ū and v (w = ū + v).
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1. There are 220 seventh-grade students. To estimate the number of students who preferred the turkey club, write the proportion who preferred the turkey club.
The correct answer to this question is 7/220. To calculate this answer, divide the number of students who preferred the turkey club (7) by the total number of seventh-grade students (220). The fraction form of this answer is 7/220, the decimal form is 0.0318, and the percentage form is 3.18%.
What is proportion?Proportion is a mathematical concept that compares two values, which can be the same or different, using division. It is used to identify the relationship between two or more values. It can also be used to compare the parts of a whole or to compare two different wholes. Proportions can be expressed as a fraction, a decimal, or a percentage.
This answer can be used to estimate the number of students who preferred the turkey club. To do this, multiply the total number of seventh-grade students (220) by the proportion (7/220). This calculation results in 7 students. In other words, approximately 7 of the 220 seventh-grade students preferred the turkey club.
The proportion 7/220 can also be used to estimate the number of students who preferred the turkey club if the total number of seventh-grade students changes. For example, if the total number of seventh-grade students increases to 250, the estimated number of students who preferred the turkey club can be calculated by multiplying the new total (250) by the proportion (7/220). This calculation results in 8 students.
Therefore, the correct answer to the question is 7/220.
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Mr. Sonny's science class is calculating the average number of blinks per minute. Jan blinks 54 times in 3 minutes. What is her blinking rate, in blinks per minute?
Answer: 18 blinks per minute
Step-by-step explanation: so for this you take 54 and divide it by three
54/3=18
blinks per minute=18
Answer:
18
Step-by-step explanation:
54\3=18
therefore Jan blinks 18 times per minutes
 If triangle ABC ~ triangle AVW, find the values of x
The value of x that satisfies the proportion is x = 7.5.
Since triangle ABC ~ triangle AVW, we know that their corresponding sides are proportional. In other words:
AB/AV = BC/VW = AC/AW
We are given that AB = 8, BC = 5, AC = 7, AV = 12, and VW = x. We need to find the value of x that satisfies the proportion above.
Using the first two ratios, we get:
AB/AV = BC/VW
8/12 = 5/x
Multiplying both sides by 12x, we get:
8x = 60
x = 7.5
Now, we can use the third ratio to check our answer:
AB/AV = AC/AW
8/12 = 7/AW
Multiplying both sides by AW, we get:
8AW = 84
AW = 10.5
We can now check that the ratios are all equal:
AB/AV = BC/VW = AC/AW
8/12 = 5/7.5 = 7/10.5
Simplifying each fraction, we get:
2/3 = 2/3 = 2/3
Therefore, the value of x that satisfies the proportion is x = 7.5.
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Perform long division on the following, please show work:
a) 33.143 / 100
b) 175.876 / 0.01
a) 0.33143; b) 17586.76
a) 33.143 / 100
33.143 ÷ 100
= 0.33143
b) 175.876 / 0.01
175.876 ÷ 0.01
= 17586.76
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15 POINTSSS!!!!!!! HELP A GIRL OUTTTTTTTT
The function that represents an exponential growth would be = f(X) = 0.3 ( 1.05) ^x. That is option C.
What is an exponential growth?An exponential growth of a function is defined as the growth that shows increase rates with respect to a given time.
There are two types of exponential functions: exponential growth and exponential decay.
The function that shows an exponential growth has the formula = f (x) = b^x, when b > 1.
From the examples given in the options above, the correct option that represents the exponential growth is f(X) = 0.3 ( 1.05) ^x. Thai so because 1.05 is greater than 1.
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A tabletop in the shape of a trapezoid has an area of 6,731 square centimeters. Its longer base measures 127 centimeters, and the shorter base is 85 centimeters. What is the height?
Answer:
[tex]\boxed{The \ height \ of \ the \ tabletop \ is \ 63.5 \ centimeters}[/tex]
Step-by-step explanation:
We are given that,
Length of the base = 127 centimeters
Width of the base = 85 centimeters
Area of the trapezoid shaped base = 6,731 square centimeters.
Since, we know,
[tex]\bold{Area \ of \ a\ trapezoid}=\frac{Length +Width}{2}\times Height[/tex]
So, we get,
[tex]6731=\frac{127+85}{2}\times Height[/tex]
i.e. [tex]6731=\frac{212}{2}\times Height[/tex]
i.e. [tex]6731=106\times Height[/tex]
i.e. [tex]Height=\frac{6731}{106}[/tex]
i.e. Height = 63.5 centimeters
Hence, the height of the tabletop is 63.5 centimeters.
PLSSSS HELP IF YOU TURLY KNOW THISSS
Answer:
You are distributing (multiplying) the three to all the terms in the parentheses.
Start:
3(x-2) = 4x + 2
Next:
3x - 6 = 4x +2
After that simplify:
-8 = x
Hope this helps!
Answer:
3x - 6 = 4x + 2 x = -8Step-by-step explanation:
Now we have to,
→ Find the required value of x.
The property we use,
→ Distributive property.
The equation is,
→ 3(x - 2) = 4x + 2
Then the value of x will be,
→ 3(x - 2) = 4x + 2
→ 3(x) - 3(2) = 4x + 2
→ 3x - 6 = 4x + 2
→ 3x - 4x = 2 + 6
→ -x = 8
→ [ x = -8 ]
Hence, the value of x is -8.
Two families attended a baseball game the first family bought three bags of popcorn in for souvenir cups which totaled $40 that second family but eight bags of popcorn in for souvenir cups which totaled $60 how much did one bag of popcorn cost
Two families attended a baseball game bought popcorns and souvenir, the cost of each pack of popcorn is $4 and cost of each pack of souvenir is $7.
An expression that depicts the relationship between two or more variables and numbers is called an equation. The provided information can be represented as, using equations:
let x be the price of popcorn and y be the price of souvenir.
First family:
[tex]3x+4y=40[/tex](equation 1)
Second family:
[tex]8x+4y=60[/tex] (equation 2)
We have a system of 2 equations with 2 variables.
3x + 4y = 40
8x + 4y = 60
We are asked for the price of 1 bag of popcorn, x, so we will eliminate y and solve for x.
from equation 2,
[tex]8x+4y=60\\4y=60-8x[/tex]
divide both sides by 4
[tex]\frac{4y}{4} =\frac{60-8x}{4} \\y=15-2x[/tex]
now putting the value of y in equation 1
[tex]3x+4(15-2x)=40\\3x+60-8x=40\\-5x=-20\\x=4\\[/tex]and [tex]y=7[/tex]
Hence, the cost of popcorn is $4 and cost of souvenir is $7.
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What are the Big Ideas or Focal Points in the mathematics
curriculum in grades 6?
The Big Ideas or Focal Points in the mathematics curriculum for grade 6 are: 1. Ratios and Proportional Relationships
2. The Number System. 3. Expressions and Equations. 4. Geometry. 5. Statistics and Probability.
1. Ratios and Proportional Relationships: Students should be able to understand and use ratios to describe relationships between quantities, and use proportional reasoning to solve problems.
2. The Number System: Students should be able to understand and use negative numbers, and perform operations with multi-digit numbers and decimals.
3. Expressions and Equations: Students should be able to write and evaluate expressions, and use equations to solve problems.
4. Geometry: Students should be able to understand and use the properties of shapes and the relationships between them.
5. Statistics and Probability: Students should be able to use data to make predictions and understand the concept of probability.
These Big Ideas or Focal Points are the key concepts that students should master in grade 6 mathematics in order to be prepared for higher-level math courses.
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The regular price of a laptop computer is $865. 0. This week, it is on sale for $812. 00 off. What is the sale price of the computer?
The sale price of the laptop computer is $53.00. We subtracted the discount amount from the original price.
Discount is a reduction in the price of a product or service that is offered to customers by a seller, usually as an incentive to increase sales or clear out inventory. It is typically expressed as a percentage of the original price.
To calculate the sale price of the laptop computer, we need to subtract the discount amount from the original price.
Discount amount = Regular price - Sale price
Discount amount = $865.00 - $812.00
Discount amount = $53.00
Therefore, the sale price of the laptop computer is $812.00.
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F. Find the domain and the range of the following functions. Write your answers in set builder form. 1) \( f(x)=4 x^{2}-2 x+1 \) 2) \( g(x)=2 x-1 \), where \( x \neq 2 \) 3) \( h(x)=2 x-\sqrt{x+1} \)
The domain and range of the functions in set builder form are:
1) Domain: \(\{x|x\in\mathbb{R}\}\), Range: \(\{y|y\in\mathbb{R}\}\)
2) Domain: \(\{x|x\in\mathbb{R}, x\neq2\}\), Range: \(\{y|y\in\mathbb{R}\}\)
3) Domain: \(\{x|x\in\mathbb{R}, x\geq-1\}\), Range: \(\{y|y\in\mathbb{R}\}\)
The domain and range of a function are the possible values of x and y, respectively, for which the function is defined. We can find the domain and range of the given functions using set builder form, which is a way of describing a set of numbers using an expression.
The function \(f(x)=4x^{2}-2x+1\) is a polynomial function, and the domain of a polynomial function is all real numbers. Therefore, the domain of this function is \(x\in\mathbb{R}\), or in set builder form, \(\{x|x\in\mathbb{R}\}\). The range of this function is also all real numbers, so the range is \(y\in\mathbb{R}\), or in set builder form, \(\{y|y\in\mathbb{R}\}\).
The function \(g(x)=2x-1\) is also a polynomial function, so the domain is all real numbers except for x=2, since the function is undefined at that point. Therefore, the domain of this function is \(x\in\mathbb{R}, x\neq2\), or in set builder form, \(\{x|x\in\mathbb{R}, x\neq2\}\). The range of this function is also all real numbers, so the range is \(y\in\mathbb{R}\), or in set builder form, \(\{y|y\in\mathbb{R}\}\).
The function \(h(x)=2x-\sqrt{x+1}\) is a radical function, and the domain of a radical function is the set of values for which the expression under the radical is greater than or equal to zero. Therefore, the domain of this function is \(x\in\mathbb{R}, x\geq-1\), or in set builder form, \(\{x|x\in\mathbb{R}, x\geq-1\}\). The range of this function is also all real numbers, so the range is \(y\in\mathbb{R}\), or in set builder form, \(\{y|y\in\mathbb{R}\}\).
In conclusion, the domain and range of the functions in set builder form are:
1) Domain: \(\{x|x\in\mathbb{R}\}\), Range: \(\{y|y\in\mathbb{R}\}\)
2) Domain: \(\{x|x\in\mathbb{R}, x\neq2\}\), Range: \(\{y|y\in\mathbb{R}\}\)
3) Domain: \(\{x|x\in\mathbb{R}, x\geq-1\}\), Range: \(\{y|y\in\mathbb{R}\}\)
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