f(x)= −2x^3 + 9x^2 + 24x − 1 on interval [ -5 , 6 ]
- extreme value theorem applies
- extreme value theorem does NOT apply
apply absolute min when x = ___
absolute min value is = ___ absolute max when x = ___ absolute max value is = ___ (enter answers rounded to the nearest tenth)
Answers
Absolute minimum value of the function = f(-1) = -26.4144
Absolute maximum value of the function = f(4) = 53
Given the function `f(x)= −2x³ + 9x² + 24x − 1` on the interval [ -5 , 6 ].
We need to determine whether the extreme value theorem applies or not.
Extreme value theorem:
If a function `f(x)` is continuous on the closed interval [a, b], then there exist two numbers c and d in the interval [a, b] such that f(c) is the maximum value and f(d) is the minimum value of the function f(x) over the interval [a, b].
If a function is continuous on a closed interval, then the absolute maximum and absolute minimum values of the function are attained at either the endpoints of the interval or at a critical point in the interval.
The given function
`f(x)= −2x³ + 9x² + 24x − 1`
is a polynomial function and polynomial functions are continuous over the entire real line.
Since the interval [-5, 6] is a closed interval, the extreme value theorem applies.
Absolute Max and Min:
The critical points of the given function `f(x)= −2x³ + 9x² + 24x − 1` can be found by taking the derivative of the function. `f(x) = -2x³ + 9x² + 24x - 1`
Differentiating w.r.t `x`, we get `f'(x) = -6x² + 18x + 24`
Setting `f'(x) = 0`, we get
`-6x² + 18x + 24 = 0`
Dividing both sides by -6, we get
`x² - 3x - 4 = 0`
Factoring, we get
`(x - 4)(x + 1) = 0`
Thus, the critical points are x = -1 and x = 4.
Both -5 and 6 are the endpoints of the interval.
We can find the maximum and minimum of `f(x)` at `x = -5, x = -1, x = 4, x = 6`.
We can summarize our observations in the table given below.
xf(x)-5-186-13.340.-1-26.4144-142.24 4 53 6 -67.142
Since the given function is a cubic function, it is continuous throughout the interval.
Hence the absolute minimum and absolute maximum values of the function are attained at either the endpoints of the interval or at a critical point in the interval.
Since the minimum value is the absolute minimum value of the function and the maximum value is the absolute maximum value of the function, the absolute minimum and absolute maximum values of the function are given as follows.
Absolute minimum value of the function = f(-1) = -26.4144
Absolute maximum value of the function = f(4) = 53
The value of `f(x)` when `x = -1` is the absolute minimum value and the value of `f(x)` when `x = 4` is the absolute maximum value.
Thus, the extreme value theorem applies to the function `f(x)= −2x³ + 9x² + 24x − 1` on the interval [ -5 , 6 ].
Absolute min is attained at `x = -1` and its value is `-26.4`
Absolute max is attained at `x = 4` and its value is `53`
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Related Questions
write a polynomial that has zeros of -3,2 and 4
Answers
\(\begin{cases} x = -3 &\implies x +3=0\\ x = 2 &\implies x -2=0\\ x = 4 &\implies x -4=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{ ( x +3 )( x -2 )( x -4 ) = \stackrel{0}{y}}\implies (x+3)(x^2-6x+8)=y \\\\\\ x^3-6x^2+8x+3x^2-18x+24=y\implies \boxed{x^3-3x^2-10x+24=y}\)
29.
After 5 years, the sum of the ages of Reema and Neha will be 30. If Reema is 4 years
younger than Neha today, then find the sum of the present ages of Reema and Neha.
years
Answers
Answer:
Sum of their present ages = 20 years
Step-by-step explanation:
Present age :
Neha's age = x years
Reemas' age = x - 4
After 5 years:
Neha's age = (x +5) years
Reemas' age = (x - 4 + 5 ) = x + 1
Sum of the ages of Reema and Neha will be 30
Therefore,
x + 5 + x + 1 = 30
x + x + 5 + 1 = 30
2x + 6 = 30
2x = 30 - 6
2x = 24
x = 24/2
x = 12
Neha's age = 12 years
Reema's age = 12 - 4 = 8 years
Sum of their present ages = 12 + 8 = 20 years
write the equation in spherical coordinates. (a) x2 + y2 + z2 = 81
Answers
The equation in spherical coordinates is:
\($\sin^2(\phi)\cos^2(\theta) + \sin^2(\phi)\sin^2(\theta) + \cos^2(\phi) = 1$\)
What is Equation in Spherical Coordinates?
A mathematical equation that is represented in terms of the spherical coordinates of a point is known as an equation in spherical coordinates. A three-dimensional coordinate system known as spherical coordinates makes use of two angles, typically represented by symbols and a radial distance (r), and a coordinate system to find points in space.
\($r^2 = 81$\)
To represent the equation in spherical coordinates, we substitute the Cartesian coordinates \($x = r\sin(\phi)\cos(\theta)$, $y = r\sin(\phi)\sin(\theta)$, and $z = r\cos(\phi)$\) into the equation. After substitution and simplification, we have:
\($r^2\sin^2(\phi)\cos^2(\theta) + r^2\sin^2(\phi)\sin^2(\theta) + r^2\cos^2(\phi) = 81$\)
Since \(r^2 = 81,\) we can substitute it into the equation:
\($81\sin^2(\phi)\cos^2(\theta) + 81\sin^2(\phi)\sin^2(\theta) + 81\cos^2(\phi) = 81$\)
Finally, we divide the equation by 81 to simplify:
\($\sin^2(\phi)\cos^2(\theta) + \sin^2(\phi)\sin^2(\theta) + \cos^2(\phi) = 1$\)
So, the equation in spherical coordinates is:
\($\sin^2(\phi)\cos^2(\theta) + \sin^2(\phi)\sin^2(\theta) + \cos^2(\phi) = 1$\)
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if a regression line is parallel to the horizontal axis of the scattergram, the slope (b) will be
Answers
If a regression line is parallel to the horizontal axis of a scattergram, it means that there is no relationship between the two variables being plotted. In this case, the slope (b) of the regression line would be zero.
When we perform a linear regression analysis, we are trying to find the best-fitting line that represents the relationship between the independent variable (x) and the dependent variable (y). The slope (b) of this line represents the rate of change between the two variables. If the regression line is parallel to the horizontal axis, it suggests that there is no change in the dependent variable for any change in the independent variable.
The general equation for a linear regression line is:
y = a + bx
Here, "a" represents the y-intercept (the value of y when x is zero) and "b" represents the slope. When the regression line is parallel to the horizontal axis, it means that the line is perfectly horizontal, and the dependent variable (y) does not change as the independent variable (x) changes.
Mathematically, this can be represented as:
y = a + 0x
y = a
In this equation, the slope (b) is zero because there is no change in the dependent variable (y) for any change in the independent variable (x). The value of y remains constant, resulting in a horizontal line parallel to the x-axis.
To further explain, when the slope (b) is zero, it indicates that there is no linear relationship between the two variables. In a scattergram, the points are spread out randomly and do not follow any specific trend or pattern. Each value of x corresponds to a single value of y, and these values do not exhibit any systematic change as x increases or decreases.
Visually, a regression line that is parallel to the horizontal axis will appear as a flat line, with all points lying on the same y-value. This indicates that the dependent variable does not depend on the independent variable and remains constant across all values of x.
In conclusion, when a regression line is parallel to the horizontal axis in a scattergram, the slope (b) of the line is zero. This indicates that there is no linear relationship between the variables being analyzed, and the dependent variable does not change as the independent variable varies. The absence of a slope suggests that the two variables are not related in a linear fashion, and the scattergram does not exhibit any pattern or trend.
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Examine the words and/or phrases below and determine the relationship among the majority of words/phrases. Choose the option which does not fit the pattern.
horn
arête
drumlin
lateral moraine
Answers
The majority of the words/phrases are related to glacial features or landforms. They represent various aspects of glacial processes and landform formation. However, "drumlin" does not fit this pattern as it specifically refers to a type of glacial landform.
Among the words/phrases provided, the majority are geological features or landforms. "Horn," "arête," "drumlin," and "lateral moraine" are all specific terms used in geology to describe different land formations.
A "horn" is a pointed mountain peak formed by the erosion of glaciers from several sides. An "arête" is a narrow ridge that separates two adjacent glacial valleys. A "drumlin" is an elongated hill formed by glacial deposition and erosion. A "lateral moraine" is a ridge of debris deposited along the sides of a glacier.
These words are related to glacial processes and landforms, specifically. They all represent different features associated with glacial erosion, deposition, or the formation of glacial landforms.
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A quadrilateral is inscribed in the circle.What is the value of x?20(6y + 12)(4x + 80°1688(6x-28)(5y + 3)18
Answers
4x + 8 + 6y + 12 + 5y + 3 + 6x - 28 = 360
10x + 11y + -5 = 360
10x + 11y = 365
Substituting the possible solutions
10(20) + 11y = 365
200 + 11y = 365
11y = 365 - 200
11y = 165
y = 165/11
y = 15
So, after doing this procedure for all the possible solutions, the only with a whole number as a result is the number 20.
So, the result is 20.
When a racer glides off a cliff, they can travel 75.2 feet. How far would the racer travel if they go over 3 cliffs?
Answers
Answer:
225.6 feet
Step-by-step explanation:
It should be pretty simple, you need to multiply 75.2 times 3. In result your answer is 225.6 feet or rather more simply (225 feet 7 13/64 inches.)
for some particular value of , when is expanded and like terms are combined, the resulting expression contains exactly terms that include the four variables , , , and each to some positive power. what is ?
Answers
Using polynomial expansion, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power is 14.
According to the question,
For some particular value of N, when (a + b + c+ d+ 1)² is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power.
Polynomial expansion :
(a + b)ⁿ = Cₙ⁰ aⁿ + Cₙⁿ⁻¹ aⁿ⁻¹ b + ....+ Cₙⁿbⁿ
\(x_{a} + x_{b} + x_{c} + x_{d} + x_{1}\) =N
\(x_{a} > 1 x_{b} > 1x_{c} > 1x_{d} > 1x_{1}\) >0
Positive integer solution:
\(x_{a} + x_{b} + x_{c} + x_{d} + x_{1}\) = N+1
Non - Negative integer solution:
\(x_{a} + x_{b} + x_{c} + x_{d} + x_{1}\) =N -4
C⁴ₙ = 1001
1001 = 7.11.13 after some trial and error we find that is N is equal to 14.
Hence, using polynomial expansion, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power is 14.
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PLSSSSSSSSSSSSSSSS I NEED HELP
Problem Solving: Use
Reasoning
1. The delivery person stopped on the 14th floor to talk to a friend. Before stopping, he had just made a delivery 4 floors above. Before that he made a delivery 6 floors below. Before that he had made a delivery 9 floors above. Before that he had made a delivery 15 floors below. On what floor did he make his first delivery?
2. Geometry The volume of a rectangular prism is 208 cm3. If the area of one end is 16 cm2, what is the length of the prism?
3. On one day, a store sold 16 boxes of rice, restocked the shelf with 22 boxes, sold 27 boxes, restocked with 30 boxes, and sold 15 boxes. There are now 21 boxes of rice on the shelf. How many boxes were on the shelf at the start of the day?
4. At the end of the day, Brooke had $138.75 in her checking account. She had made a deposit of $115.07 and written checks totaling $176.94. How much did she have in her checking account at the beginning of the day?
A –$76.88
B $76.88
C $200.62
D $430.76
5. Writing to Explain The football team gained 7 yards, gained 4 yards, lost 5 yards, gained 21 yards, lost 2 yards, and gained 4 yards to their 43 yard line. Explain how you solved this problem. Then find the yard line where the team began.
Practice 7-6
Answers
Answer:
1. 6th Floor
2. 13 cm
3. 15 boxes
4. $76.88
5. 14
Step-by-step explanation:
1. 14 + 4 = 18. 18 - 6 = 12. 12 + 9 = 21. 21 - 15 = 6
2. 208 ÷ 16 = 13
3. 21 - 15 = 6. 6 + 30 = 36. 36 - 27 = 9. 9 + 22 = 31. 31 - 16 = 15
4. 138.75 + 115.07 = 253.82. 253.82 - 176.94 = 76.88
5. ? + 7 + 4 - 5 + 21 - 2 + 4 = 43. First you need to combine all of it:
? + 29 = 43. 43 - 29 = ?
43 - 29 = 14
? = 14
Write a function for the following table
X. Y
-1. -4
0. -2
1. -1
2. -0.5
Please help I only have 20 minutes
Answers
Answer: x+1 = y/2
Step-by-step explanation:
X is increasing by 1
(Adding 1 to each number)
For Y, divide it by 2
-4/2 = -2
-2/2 = -1, etc.
When a solution to the linear programming problem satisfies all the constraints, including the nonnegativity conditions, it is considered Feasible O Semi-feasible Infeasible O Unbounded 1 points Save Answer By reviewing and evaluating up Solution, we have a. taken the final step in the Decision-Making proces
b. taken the first step in the Decision Making proces c. considered multiple criterias to evaluate the alterr
d. proposed all possible alternatives for the problem
Answers
Option C. When a solution to the linear programming problem satisfies all the constraints, including the nonnegativity conditions, it is considered Feasible.
In linear programming, a feasible solution is a solution that satisfies all of the constraints in the problem, including the nonnegativity conditions. In other words, a feasible solution is a solution that is possible and meets all of the requirements of the problem.
On the other hand, an infeasible solution is a solution that violates one or more of the constraints or the nonnegativity conditions. An unbounded solution is a solution where the objective function can be increased or decreased indefinitely, without violating any of the constraints.
Therefore, when a solution to the linear programming problem satisfies all the constraints, including the nonnegativity conditions, it is considered feasible.
Option A, B, C, and D are not directly related to the concept of feasibility in linear programming.
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When a solution to the linear programming problem satisfies all the constraints, including the nonnegativity conditions, it is considered Feasible O Semi-feasible Infeasible O Unbounded 1 points Save Answer By reviewing and evaluating up Solution, we have
A. taken the final step in the Decision-Making process
B. taken the first step in the Decision Making process
C. considered multiple criterias to evaluate the alter
D. proposed all possible alternatives for the problem
In VWX and PQR similar? How do you know?
Answers
The collection of whole numbers is an example of a set. OA. True O B. False
Answers
Answer:
It is false
Step-by-step explanation:
Click the photo to solve the photo
Answers
Answer:
A=2
B=4
C=6
D=5
E=7
F=8
G=3
H=1
Step-by-step explanation:
explanation is in the picture!
Heng was trying to factor 10 x 2 + 5 x 10x 2 +5x10, x, squared, plus, 5, x. She found that the greatest common factor of these terms was 5 x 5x5, x and made an area model:
Answers
Full question:
Heng was trying to factor 10x²+5x. She found that the greatest common factor of these terms was 5x and made an area model: What is the width of Heng's area model?
Answer:
The width of the area model is 2x + 1
Step-by-step explanation:
Given
Expression: 10x² + 5x
Factor: 5x
Required
Width of the Area Model
To solve this, I'll assume the area model is Length * Width
Provided that we're to solve for the width of the model.
This implies that; Length = 5x
Area = Length * Width
And
Area = 10x² + 5x
Equate these two
Length * Width = 10x² + 5x
Factorize express on the right hand side
Length * Width = 5x(2x + 1)
Substitute 5x for Length
5x * Width = 5x(2x + 1)
Divide both sides by 5x
Width = 2x + 1
Hence, the width of the area model is 2x + 1
Whats the y-intercept of -2x - y=3
Answers
m is the slope and b is the y intercept.
-2x-y=3
y=-2x-3
So the b, or the y intercept is -3
A quadrilateral that is NOT a parallelogram with two diagonals that are congruent.
Answers
Answer:
isosceles trapezoid
Step-by-step explanation:
isosceles trapezoid
Given f(x) = -x + 2, find f(0)
Answers
Step-by-step explanation:
Using the f(x) defined in the question,
f(0) = -(0) + 2 = 2.
Please help me it’s urgent
Answers
Answer:
Picture 1:( x-2.5)²+(y+5.5)²=225
Picture 2:( x-4)²+(y-1)²=25
Step-by-step explanation:
Use the rule: ( x-p)²+(y-q)² =r
( p,q)is the coordinate of the center of the circle
r is the radius of the circle
x^2 + 5x + 6
enter your next step here:
Answers
then you can turn this into (x+2)(x+3)
4. Calculate the values for the ASN curves for the single sampling plan \( n=80, c=3 \) and the equally effective double sampling plan \( n_{1}=50, c_{1}=1, r_{1}=4, n_{2}=50, c_{2}=4 \), and \( r_{2}
Answers
Single Sampling Plan: AQL = 0, LTPD = 3.41, AOQ = 1.79 Double Sampling Plan: AQL = 0, LTPD = 2.72, AOQ = 1.48
The values for the ASN (Average Sample Number) curves for the given single sampling plan and double sampling plan are:
Single Sampling Plan (n=80, c=3):
ASN curve values: AQL = 0, LTPD = 3.41, AOQ = 1.79
Double Sampling Plan (n1=50, c1=1, r1=4, n2=50, c2=4, r2):
ASN curve values: AQL = 0, LTPD = 2.72, AOQ = 1.48
The ASN curves provide information about the performance of a sampling plan by plotting the average sample number (ASN) against various acceptance quality levels (AQL). The AQL represents the maximum acceptable defect rate, while the LTPD (Lot Tolerance Percent Defective) represents the maximum defect rate that the consumer is willing to tolerate.
For the single sampling plan, the values n=80 (sample size) and c=3 (acceptance number) are used to calculate the ASN curve. The AQL is 0, meaning no defects are allowed, while the LTPD is 3.41. The Average Outgoing Quality (AOQ) is 1.79, representing the average quality level of outgoing lots.
For the equally effective double sampling plan, the values n1=50, c1=1, r1=4, n2=50, c2=4, and r2 are used. The AQL and LTPD values are the same as in the single sampling plan. The AOQ is 1.48, indicating the average quality level of outgoing lots in this double sampling plan.
These ASN curve values provide insights into the expected performance of the sampling plans in terms of lot acceptance and outgoing quality.
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Al Medina, D.D.S, opened an incorporated dental practice on January 1, 2022. During the first month of operations, the following transactions occurred.
Performed services for patients who had dental plan insurance. At January 31, $760 of such services was completed but not yet billed to the insurance companies.
Utility expenses incurred but not paid prior to January 31 totaled $450.
Purchased dental equipment on January 1 for $80,000, paying $20,000 in cash and signing a $60,000, 3-year note payable (interest is paid each December 31). The equipment depreciates $400 per month. Interest is $500 per month
Purchased a 1-year malpractice insurance policy on January 1 for $24,000.
Purchased $1,750 of dental supplies (recorded as increase to Supplies). On January 31, determined that $550 of supplies were on hand.
Prepare the adjusting entries on January 31. Account titles are Accumulated Depreciation- Equipment, Depreciation Expense, Service Revenue, Accounts Receivable, Insurance Expense, Interest Expense, Interest Payable, Prepaid Insurance, Supplies, Supplies Expense, Utilities Expense, and Accounts Payable. (If no entry is required, select "No Entry for the account titles and enter O for the amounts. Credit account titles are automatically indented when the amount is entered. Do not indent manually.
Answers
The entry to record the services provided to patients with dental plan insurance but not yet billed is: Debit: Accounts Receivable - Dental Plan Insurance $760 ,Credit: Dental Services Revenue $760.
How to prepare the journal entry?To prepare the adjusting entries on January 31 for Al Medina, D.D.S, we need to analyze the transactions and determine the amounts that need to be recorded as expenses, prepaid expenses, and accrued expenses. Based on the information given, the following adjusting entries are required:
To record the services provided to patients with dental plan insurance but not yet billed:
Debit: Accounts Receivable - Dental Plan Insurance $760
Credit: Dental Services Revenue $760
To record the utility expense incurred but not yet paid:
Debit: Utilities Expense $450
Credit: Utilities Payable $450
To record the depreciation expense for the dental equipment:
Debit: Depreciation Expense $400
Credit: Accumulated Depreciation - Dental Equipment $400
To record the interest expense on the note payable:
Debit: Interest Expense $500
Credit: Interest Payable $500
To record the portion of the malpractice insurance policy that has been used up in January:
Debit: Insurance Expense $2,000 ($24,000 ÷ 12 months)
Credit: Prepaid Insurance $2,000
To adjust the Supplies account for the supplies used up in January:
Debit: Supplies Expense $1,200 ($1,750 - $550)
Credit: Supplies $1,200
Therefore after posting these adjusting entries, the financial statements will reflect the correct balances in the accounts and the expenses and revenues for the month of January will be accurately reported.
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List all the permutations of four objects m, I, n, and k taken two at a time without repetition. What is P? List all the permutations of four objects m, I, n, and k taken two at a time without repetition. Choose the correct answer below. O A. m, mn, mk, In, lk, nk OB. ml, mn, mk, Im, in, lk, nm, nl, nk, km, kl, kn OC. m. I, n,k OD. mm, ml, mn, mk, II, In, lk, nn, nk, kk What is P2?
Answers
The permutations of four objects m, I, n, and k taken two at a time without repetition are: mn, mk, mi, nm, nk, ni, km, kn, ki, in, im, ik.
P is the total number of permutations, which is equal to 12.
The correct answer is OB, which lists all 12 permutations.
P2 is the number of permutations taken two at a time with repetition allowed. This means that we can repeat the same object in a permutation.
There are 16 possible permutations with repetition allowed: mm, ml, mn, mk, ll, ln, lk, nn, nk, kk, ii, ik, nn, ni, nk.
Therefore, P2 is equal to 16.
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g (1) = type your answer.
g (-12)= type your answer
Answers
The numeric values of the graphed function are given as follows:
g(1) = 4.g(-12) = -10.How to obtain the numeric values from the graph of a function?To obtain the numeric values of a function from its graph, you can use the following steps:
Determine the x-value of the point on the graph where you want to find the corresponding y-value.Locate the point on the y-axis that corresponds to the x-value you found in step 1.Read the y-value off the y-axis at the point you located in step 2. This is the numeric value of the function at the x-value you found in step 1.When x = 1, the point with a closed circle is at y = 4, hence the numeric value is given as follows:
g(1) = 4.
When x = -12, we have that y = -10, hence the numeric value is given as follows:
g(-12) = -10.
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the exponent of 2√9 is
Answers
Answer:
the exponent is 6............
Q. Find missing angle
a.60
b.36
c.30
d.41
e.49
Answers
Answer:
b or c
Step-by-step explanation:
Lacy uses 1 1/2 cups of flour and 1/2 cup of applesauce for one batch of muffins. She makes 2 1/2 batches of muffins. How many
cups of flour does Lacy use in all?
Answers
If Lacy is making 2 1/2 batches them you do the math like this :
1 1/2 + 1 1/2 + 3/4 = 3 1/4
this is because you will need the full 1 1/2 cups for the first 2 batches then you’ll have to divide the 1 1/2 by 2 to half the recipe.
Hope this helps <3
Please help ASAPPPPPO
Answers
Answer:
x=all real numbers
y=all real numbers
Step-by-step explanation:
x=-5y-1
-3(-5y-1)-15y=3
15y+3-15y=3
3=3
therefore the answer is all real numbers
help!!! an experiment consists of randomly choosing a colored card from a box. use the results in the table to find the experimental probability of each event. simplify all answers. red : 7, yellow: 12, orange: 8, white: 13
Answers
Answer:
chance of:
red 7/40 = 17.5%
yellow 3/10 = 30%
orange 1/5 = 20%
white = 13/40 = 32.5%
Step-by-step explanation:
7+12+8+13 = 40 total cards
7 red/ 40 possible
12 yellow/40 possible = 3/10
8 orange/40 possible = 1/5
13 white/40possible
Using it's concept, it is found that the experimental probability of each outcome is given by:
Red: 0.175 = 17.5%.Yellow: 0.3 = 30%.Orange: 0.2 = 20%.White: 0.325 = 32.5%.What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
In an experimental probability, these numbers are taken from previous outcomes.
In this problem, there are 40 cards.
7 are red, hence P(red) = 7/40 = 0.175 = 17.5%.12 are yellow, hence P(yellow) = 12/40 = 0.3 = 30%.8 are orange, hence P(orange) = 8/40 = 0.2 = 20%.13 are white, hence P(white) = 13/40 = 0.325 = 32.5%.More can be learned about probabilities at https://brainly.com/question/14398287
A carpenter has been hired to construct a sign for a pet grooming business. The plans for the sign call for an elliptical shape with an eccentricity of 0.60 and a length of 36 inches. What is the maximum height of the sign?
Answers
Answer:
The maximum height of the sign is 3.6 inches.
Step-by-step explanation:
To obtain the maximum height of the sign, we need to understand the relationship between the eccentricity and the dimensions of an ellipse.
The eccentricity of an ellipse measures how elongated or stretched the ellipse is. It is defined as the ratio of the distance between the foci of the ellipse to the length of the major axis.In this case, the eccentricity is given as 0.60, which means the distance between the foci is 0.60 times the length of the major axis.
The length of the major axis is 36 inches, we can calculate the distance between the foci:
Distance between foci = eccentricity * length of major axis
Distance between foci = 0.60 * 36 inches
Distance between foci = 21.6 inches
Now, to obtain the maximum height of the sign, we need to consider that the distance from the center of the ellipse to the highest point (vertex) is equal to the distance from the center to one focus.
The maximum height of the sign is half the length of the minor axis, and it can be calculated using the formula:
Height = 0.5 * (length of major axis - 2 * distance between foci)
Height = 0.5 * (36 inches - 2 * 21.6 inches)
Height = 0.5 * (36 inches - 43.2 inches)
Height = 0.5 * (-7.2 inches)
Height = -3.6 inches
However, negative height does not make sense in this context, so we take the absolute value of the height:
Absolute Height = |-3.6 inches
Absolute Height = 3.6 inches
Therefore, the maximum height of the sign is 3.6 inches.
Learn more about height here, https://brainly.com/question/12446886
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