The absolute maximum value on (0, ∞) for f(x) = 4x - 2x ln x is approximately 2e (5.436), which occurs at x = e.
To find the absolute maximum value on (0, ∞) for the function f(x) = 4x - 2x ln x, we need to follow these steps:
1. Determine the critical points of the function by finding its first derivative and setting it equal to zero.
2. Check the critical points for the maximum value.
3. Verify the behavior at the boundary of the interval (0, ∞).
Step 1: Find the first derivative of f(x).
f(x) = 4x - 2x ln x
f'(x) = d/dx (4x) - d/dx (2x ln x)
Using the product and constant rules, we get:
f'(x) = 4 - 2(ln x + 1)
Step 2: Set the first derivative equal to zero to find the critical points.
4 - 2(ln x + 1) = 0
2(ln x + 1) = 4
ln x + 1 = 2
ln x = 1
Solving for x:
x = e^1
x = e (approximately 2.718)
Step 3: Check the behavior at the boundaries.
As x approaches 0 from the right, ln x approaches negative infinity, making the term -2x ln x approach infinity. Since the interval is (0, ∞), we only need to consider the behavior as x approaches ∞. As x goes to infinity, both 4x and -2x ln x will also go to infinity. However, -2x ln x will increase at a slower rate compared to 4x, so f(x) will approach infinity.
Now, we need to check the value of f(x) at the critical point x = e:
f(e) = 4e - 2e ln e
f(e) = 4e - 2e(1)
f(e) = 2e (approximately 5.436)
Thus, the absolute maximum value on (0, ∞) for f(x) = 4x - 2x ln x is approximately 5.436, which occurs at x = e.
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COMPARE BY USING <,>,=,<=,>=
The product of the middle two sums is greater than or equal to the product of the least and the greatest of the sums.
How can the two products be compared?To compare two products, we need to compare the values of the products using the comparison operators (<, >, <=, >=, or =).
We can start by finding the sums of the original numbers (0, 1, 2, 3):
Sum of the original numbers = 0 + 1 + 2 + 3 = 6
Now, we add the number k to each of the numbers:
Sum of the new numbers = (0 + k) + (1 + k) + (2 + k) + (3 + k)
= (0 + 1 + 2 + 3) + 4k
= 6 + 4k
So, the new sums range from 6 + 4k (the smallest) to 9 + 4k (the largest).
The product of the least and the greatest of the sums is:
(6 + 4k) × (9 + 4k) = 54 + 60k + 16k^2
The product of the middle two sums is:
(7 + 4k) × (8 + 4k) = 56 + 60k + 16k^2
Comparing the two products using the comparison operators:
54 + 60k + 16k^2 < 56 + 60k + 16k^2 (since 54 < 56)
or
54 + 60k + 16k^2 <= 56 + 60k + 16k^2 (since the products are equal when k=0)
In conclusion, the product of the middle two sums is greater than or equal to the product of the least and the greatest of the sums.
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Jill has $1275.00 in her savings account. When she opened her account, she had $300.
Every week she deposited $75.00. How many weeks did it take to earn $1275.00?
Answer:
It took 13 weeks for Jill to get 1275.00 in her savings account
Step-by-step explanation:First off you subtract 1275-300=975.After that you will divide 975 from 75 975÷75=13.So 13 is your final answer.
A cross section of the cylinder with the cone removed is a ring. To find the area of the ring, find the area of the outer circle and of the inner circle. Then subtract the area of the inner circle from the outer circle.
The area of the outer circle and of the inner circle is : πr² and, πx².
Then subtract the area of the inner circle from the outer circle is :
π (r - x) (r+x)
Here, we have,
Given:
Let the radius of outer circle i.e CA be r
Let the radius of inner circle i.e CB be x
The diagram is given below as attachment.
We know that,
area of circle = πR²
Then subtract the area of the inner circle from the outer circle is:
The area of the outer circle - area of the inner circle
= πr² - πx²
= π (r² - x²)
=π (r - x) (r+x)
So, we get
Then subtract the area of the inner circle from the outer circle is:
π (r - x) (r+x)
Hence, The area of the outer circle and of the inner circle is : πr² and, πx².
Then subtract the area of the inner circle from the outer circle is :
π (r - x) (r+x)
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Consider the vector field.
F(x, y, z) =
5ex sin(y), 9ey sin(z), 2ez
sin(x)
(a) Find the curl of the vector field.
curl F =
(b) Find the divergence of the vector field.
div F =
(a) To find the curl of the vector field F(x, y, z), we first find its component functions:
F(x, y, z) = (5ex sin(y), 9ey sin(z), 2ez sin(x))
Then, we use the formula for the curl of a vector field:
curl F = (∂Fz/∂y - ∂Fy/∂z, ∂Fx/∂z - ∂Fz/∂x, ∂Fy/∂x - ∂Fx/∂y)
Plugging in the component functions of F(x, y, z), we get:
curl F = (2ez cos(x), -5ex cos(y), 9ey cos(z))
(b) To find the divergence of the vector field F(x, y, z), we use the formula for the divergence of a vector field:
div F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z
Plugging in the component functions of F(x, y, z), we get:
div F = 5e^x sin(y) + 9e^y sin(z) + 2e^z sin(x)
(a) To find the curl of the vector field F(x, y, z) = (5e^x sin(y), 9e^y sin(z), 2e^z sin(x)), we need to compute the cross product of the del operator (∇) and F:
curl F = ∇ x F
curl F = ( (∂/∂y)(2e^z sin(x)) - (∂/∂z)(9e^y sin(z)), (∂/∂z)(5e^x sin(y)) - (∂/∂x)(2e^z sin(x)), (∂/∂x)(9e^y sin(z)) - (∂/∂y)(5e^x sin(y)) )
After computing the partial derivatives, we get:
curl F = ( 0, 5e^x cos(y) - 2e^z cos(x), 9e^y cos(z) - 5e^x cos(y) )
(b) To find the divergence of the vector field F(x, y, z), we need to compute the dot product of the del operator (∇) and F:
div F = ∇ ⋅ F
div F = (∂/∂x)(5e^x sin(y)) + (∂/∂y)(9e^y sin(z)) + (∂/∂z)(2e^z sin(x))
After computing the partial derivatives, we get:
div F = 5e^x sin(y) + 9e^y sin(z) + 2e^z sin(x)
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What is the cost of 0. 7 kg of apples, if 1 kg of apples cost 89. 50
In the proportion, the cost of 0.7kg apple is Rs.62.65 .
What is proportion?
A percentage is created when two ratios are equal to one another. We write proportions to construct equivalent ratios and to resolve unclear values. a comparison of two integers and their proportions. According to the law of proportion, two sets of given numbers are said to be directly proportional to one another if they grow or shrink in the same ratio.
Here cost of 1kg apples = 89.50
Then cost of 0.7 kg apple = x.
Now using proportion,
=> 1 = 89.50
0.7= x
=> x = [tex]\frac{89.50\times0.7}{1}[/tex]
=> x = 62.65.
Hence the cost of 0.7kg apple is Rs.62.65 .
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Subtract 1/4- 5/14 = type an integer or fraction
Answer:
-3/28
Step-by-step explanation:
In order to subtract fractions you first have to get a common denominator. The common denominator between 4 and 14 is 28 since 4(7) = 28 and 14(2) = 28. Then multiply the top and bottom of (1/4) by (7/7) and (5/14) by (2/2). This gives a common denominator and since you multiply by a fraction equal to 1, it doesn't change the value. Therefore, we get 7/28 - 10/28 = -3/28.
Tara earns $8. 50 an hour (after taxes) at the pizza place. She is scheduled to work four hours this afternoon. However, her friend Kayla called and asked her if she wants to go to the movie. A ticket to the movie costs $9. 50. In addition, she always spends about $7 on snacks
The following statement which are true are Kayla's opportunity cost to go to the movie is $9.50 and Tara's total cost of attending the movie is $50.5, option B, D.
Tara earns $8.50 per hour
She is scheduled to work for 4 hours
Total earnings=$8.50 × 4
=$34
Tara's opportunity cost of attending the movie instead of working is $34
Since, a ticket cost $9.50
And she always spend about $7 on snacks
Tara's total cost of going to the movies = opportunity cost of attending the movie + cost of tickets + cost of snacks
=$34 + $9.50 + $7
=$50.5
Opportunity cost refers to the cost of satisfying a want at the expense of another want. It can also be called REAL COST or TRUE COST.
Therefore,
2. Kayla's opportunity cost to go to the movie is $9.50
4. Tara's total cost of attending the movie is $50.5
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Complete question:
Tara earns $8.50 an hour (after taxes) at the pizza place. She is scheduled to work four hours this afternoon. However, her friend Kayla called and asked her if she wants to go to the movie. A ticket to the movie costs $9.50. In addition, she always spends about $7 on snacks.
Which of the following statement are true? Select all that apply.
1. Tara's opportunity cost if she goes to work instead of the movie is $34.
2. Kayla's opportunity cost to go to the movie is $9.50.
3. There is no opportunity cost for Tara to go to the movie.
4. Tara's total cost of attending the movie is $50.5
5. Tara's opportunity cost if she goes to the movie instead of working is $34
A boy cycles 5km from his home to school and 8km from his home to the market. The chief's camp is closer to the boys home than the market but further than the school. Write a compound inequality to show the distance from the boys home to the chief's camp.
A compound inequality to show the distance from the boys home to the chief's camp is 5 < d < 8
How to explain the inequalityThe distance from the boy's home to the school is 5km.
The distance from the boy's home to the market is 8km.
The chief's camp is closer to the boy's home than the market, but further than the school
The chief's camp is closer to the boy's home than the market, so the distance from the boy's home to the chief's camp is less than 8km.
Putting these together, we can write a compound inequality to show the possible distances d from the boy's home to the chief's camp:
distance from home to school < distance < home to maket
5 < d < 8
The inequality is 5 < d < 8.
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Fred goes to Old Navy and buys two pairs of jeans for $17. 99 each, three shirts at $5. 99 each, and a pack of socks for $3. He has a coupon for 25% off his entire purchase. Sales tax is 6. 5%. What is the total cost after the discount and tax?
A: $43. 07
B: $39. 95
C: $45. 49
D: $60. 65
The total cost after discount and tax is $46.28.
To find the total cost after the discount and tax, we need to first find the subtotal before tax.
Fred bought two pairs of jeans for $17.99 each, so the cost of the jeans is $17.99 x 2 = $35.98.
He also bought three shirts at $5.99 each, so the cost of the shirts is $5.99 x 3 = $17.97.
The pack of socks costs $3.
The subtotal before discount is $35.98 + $17.97 + $3 = $57.95.
With the 25% off coupon, Fred gets a discount of $57.95 x 0.25 = $14.49.
The new subtotal after discount is $57.95 - $14.49 = $43.46.
Finally, we need to add the 6.5% sales tax.
The sales tax is $43.46 x 0.065 = $2.82.
The total cost after discount and tax is $43.46 + $2.82 = $46.28.
Therefore, the closest answer is C: $45.49.
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A hotel offers two activity packages. One costs $192 and includes 3h of horseback riding and 2h of parasailing. The second costs $213 and includes 2h of horseback riding and 3h of parasailing. What is the cost for 1h of each activity?
Answer:
Step-by-step explanation:
Step-by-step explanation:
let's assumed that
x = 1h of horseback
y = 1h of parasailing
3h of horseback = 3x
2h of parasailing = 2y
and
2h of horseback = 2x
3h of parasailing = 3y
if
3x + 2y = 192
2x + 3y = 213
to find y we have to remove the x
(3x + 2y = 192) × 2
(2x + 3y = 213) × 3
6x + 4y = 384
6x + 9y = 639
___________ -
-5y = -255
y = 51
substitute y to any equation to find x
3x + 2y = 192
3x + 2(51) = 192
3x + 102 = 192
3x = 192 - 102
3x = 90
x = 30
so the answers are 1h of horseback = $30 and 1h of parasailing = $51
#CMIIWAnswer Immeditely Please
Answer:
it's ither 1 2 or 4 I'm still thinking but I'm pretty sure it's one of those answers
Fill in the boxes 3(n+7) = (3)( ) + (3)( ) = +
the final result will be 3n + 21
what is ditribustion propety?
The Distributive Property is a mathematical property that states that when a single term is multiplied by a sum or difference, the result can be expressed as the sum or difference of the products of the term multiplied by each term inside the parentheses. It is often used to simplify expressions and equations in algebra. 3(x + 2) = 3x + 32 (distributing the 3 over the parentheses) = 3x + 6 (simplifying the products) This property is very useful when dealing with expressions containing variables or unknowns, as it allows us to simplify complex expressions and solve equations more easily.
using distribution propety
3(n + 7)
= (3)(n) + (3)(7)
= 3n + 21
Hence the final result will be 3n + 21
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Solve the following inequality for r. Write your answer in simplest form. -5r - 2(-10r - 9)<=-6r + 8 - 9
Distribute on the left side:-5r + 20r + 18 ≤ -6r - 1
Combine like term on the left side:15r + 18 ≤ -6r -1 (Add 6 to both sides)21r + 18 ≤ -1 (Subtract 18 from both sides)21r ≤ -1921r/21 ≤ -19/21 (Divide by 21)Get Solution r ≤ -19/21Solution:r ≤ -19/21
A negatively charged balloon moves close to another balloon. They then repel each other. What can be said about the other balloon? (2 points)
A: Both balloons have a positive charge.
B: It has a negative charge.
C: The balloon is uncharged.
D: There is a positive charge.
The repulsion between two negatively charged objects is an indication that the other object must be negatively charged. Thus, the correct answer is B: It has a negative charge.
This is due to the fact that like charges repel each other, while opposite charges attract each other. In this case, the negatively charged balloon repels the other balloon, indicating that the other balloon is also negatively charged. So, the correct answer is B).
The other options are incorrect. Option A is incorrect because both balloons cannot be positively charged as they would attract each other, not repel. Option C is incorrect because an uncharged object would not repel a negatively charged object. Option D is incorrect because a positively charged object would attract the negatively charged balloon, not repel it.
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which angle is adjacent to TOU
C C
A student believes that a certain number cube is unfair and is more likely to land with a six facing up. The student rolls
the number cube 45 times and the cube lands with a six facing up 12 times. Assuming the conditions for inference
have been met, what is the 99% confidence interval for the true proportion of times the number cube would land with a
six facing up?
0. 27 2. 58
0. 221-0. 27)
45
0. 7342. 33
0. 731-0. 73)
45
0. 27 2. 33
0. 271 -0. 20)
45
0. 73 +2. 58
0. 73(10. 73)
45
Mix
Save and Exit
The answer is option B: (0.221-0.27).
Using the formula for a confidence interval for a proportion:
p± z*√(p(1-p)/n)
where p is the sample proportion (12/45 = 0.267), z* is the z-score for the desired confidence level (99% corresponds to a z-score of 2.576), and n is the sample size (45).
Substituting the values, we get:
0.267 ± 2.576*√(0.267(1-0.267)/45)
which simplifies to:
0.267 ± 0.195
Therefore, the 99% confidence interval for the true proportion of times the number cube would land with a six facing up is (0.072, 0.462).
So the answer is option B: (0.221-0.27).
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Help please! solve problem in the photo
Check the picture below.
[tex]6^2=[4+(2x+5)](4)\implies 36=(9+2x)(4)\implies \cfrac{36}{4}=9+2x \\\\\\ 9=9+2x\implies 0=2x\implies \cfrac{0}{2}=x\implies 0=x~\hfill \stackrel{ 2(0)+5 }{{\Large \begin{array}{llll} GH=5 \end{array}}}[/tex]
Answer this question please
This question can be answered as follows:
-4 + 12 = 8
How to answer the questionTo answer this question, we can begin by searching for the number whose addition to -4 will yield 8 and the answer is 12. To determine the answer, we can first begin by assigning the figure x to the unknown variable. Thus, we will have:
-4 + x = 8
We collect like terms:
x = 8 + 4
= 12
So, the number that when added to -4 will yield 8, is 12. So, to find a missing number, we assign it a variable and equate to the specified end result.
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A teacher asked three different students to write the conditions that would result in a triangle. Which of the following students listed conditions that would result in more than one triangle?
The condition that will result in more than one triangle is C. Student III.
How the conditions will result in more than one triangleThe conditions listed by the third student will result in more than one triangle because we are given all three angles. As a rule in math, some conditions will determine if there is more than one triangle. One of them is this:
Rule 1:
If all three angles of the triangle are given and they all add up to exactly 180°, it is possible to get more than one triangle. In the third option, we are given angles 62°, 36°, and 82°, so different triangles can be constructed. Also, they all add up to give 180°. The condition is satisfied.
Rule 2:
Also, as a rule, if we have two angles that do not add up to 180° and one side, then only one unique triangle can be obtained. This is the case for student A who is given angles A and B and a side length of 5cm. (ASA)
Rule 3:
Student 2 will also produce a unique triangle because there are three sides that meet the triangle inequality theorem. (SSS)
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Jane completed 8 homework problems in class. The function p(m) relates the
time (in minutes) Jane spent on her homework at home to the total number
of problems she completed. The input is the number of minutes worked. The
output is the number of problems completed.
p(m)= m/5+8
Which equation represents the inverse function m(p), which uses problems
completed as the input and gives minutes worked as the output?
Answer:
Step-by-step explanation:
5p-40
Larijah is creating a circular board game with a spinner with four regions that players use to determine what happens on their turns. she wants to meet these requirements: - exactly a quarter of the circle should contain the ""lose a turn"" region. - ""move one space"" should be three times the angle as ""move two spaces"". - ""move two spaces"" should be twice the angle as ""trade places with any opponent"". what is the measure of the ""trade places with any opponent"" region?
The measure of the trade places with any opponent region is 30°
Let the measure of the trade places with any opponent = x
Move two space is twice the angle as trade places with any opponent
Move two space = 2x
Move one space is three times the angle as moving two spaces
Move one space = 3(2x)
Move one space = 6x
Lose your turn is a quarter
Lose your turn = 90°
The sum of a complete angle = 360°
90 + x + 2x + 6x = 360°
9x = 360 - 90
9x = 270
x = 30
The measure of the trade places with any opponent region is 30°
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If a data set has many very large data values, then the mean wil be
the median
a larger than
b. the same as
c. equal to
d. smaller than
A data set has many very large data values, then the mean will be larger than the median. The correct option is a.
In a dataset with large values, the mean is influenced by outliers and extreme values, whereas the median is not. The median is the value that divides the data into two equal halves, while the mean is calculated by summing up all values and dividing by the total number of values.
Thus, in a skewed dataset with large values, the mean tends to be pulled towards the larger values, resulting in a larger mean than the median. This effect is more pronounced in datasets with a high variance. Therefore, option (a) is the correct answer.
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if y=x^2+2x-5 for -3<=x<=3 then which of the following sets is the range of y
The range of the equation y = x² + 2x -5 where -3 ≤ x ≤ 3 is [-3,10]
The function is y = x² + 2x -5
Domain of x is [-3,3]
By putting the value of x = -3
y = (-3)² + 2(-3) - 5
y = 9 - 6 -5
y = -2
Minimum possible value is -2
And by putting the value of x = 3
y = 3² + 2(3) -5
y = 9 + 6 -5
y = 10
Maximum possible value is 10
The value of y ranges between -2 ≤ y ≤ 10
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The question is incomplete the complete question is :
If y = = x² + 2x -5 for -3 ≤ x ≤ 3 then what will be the range of y ?
Can you help me with this question step by step.
Answer:
32
Step-by-step explanation:
There are 28 full shaded squares
There are 8 half squares
2 half squares make up 1 full square
So 28 + 4 = 32
Area is 32 units
Esteban has `3` kittens. According to the vet, each kitten is born with blue eyes and there is a `50\%` chance of it changing color once they reach three months.
Esteban decides to run a simulation using `3` coins, where heads represent the eyes changing color.
The table shows the results of his simulation.
Estimate the probability that at least one of Esteban’s kittens will still have blue eyes at three months old
There is an estimated probability of 0.875 (or 87.5%) that at least one of Esteban's kittens will still have blue eyes at three months old.
Let's use the complement rule to estimate the probability that at least one kitten will still have blue eyes at three months old:
P(at least one kitten with blue eyes) = 1 - P(no kitten with blue eyes)
To estimate P(no kitten with blue eyes), we can use the results of Esteban's simulation. We can see from the table that all three coins came up tails (representing no color change) in row 8, so that's one outcome where no kitten has blue eyes. There are a total of 2^3 = 8 possible outcomes, so the estimated probability of no kitten having blue eyes is:
P(no kitten with blue eyes) = 1/8 = 0.125
Therefore, the estimated probability that at least one kitten will still have blue eyes at three months old is:
P(at least one kitten with blue eyes) = 1 - P(no kitten with blue eyes) = 1 - 0.125 = 0.875
So, there is an estimated probability of 0.875 (or 87.5%) that at least one of Esteban's kittens will still have blue eyes at three months old.
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The area of the region under the curve of a function f(x)= ax+b on the interval [0,4] is 16 square units. (A,b) ≠
There are infinitely many solutions to this equation. For example, one possible solution is a = 2, b = 0. Another possible solution is a = 1, b = 2.
How to find the area?To find the area, we need to use the definite integral formula to calculate the area under the curve:
∫[0,4] f(x) dx = ∫[0,4] (ax + b) dx = 1/2 * a * x² + b * x |[0,4]
Substituting the limits of integration, we get:
1/2 * a * 4² + b * 4 - (1/2 * a * 0² + b * 0) = 16
Simplifying, we get:
8a + 4b = 16
Dividing by 4, we get:
2a + b = 4
Since (a,b) ≠ (0,0), there are infinitely many solutions to this equation. For example, one possible solution is a = 2, b = 0. Another possible solution is a = 1, b = 2.
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Find the value of X of the circle
The circle's x value will continue to be the same as the arc ST. So, x has a value of 40 degrees.
What is circle?A circle's centre is the point from which all of the distances to the other points on the circle are equal. The radius of the circle is the measurement at issue.
Every point on a circle is a geometric shape that is equally separated from the centre.
The location of any point that is evenly separated from the fixed point known as the circle's centre can also be characterised as it.
The distance from any point on a circle to the centre is known as the radius of the circle.
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Review Questions
1. A washer and a dryer cost $1004 combined. The washer costs $54 more than the dryer. What is the cost of the dryer?
Please use your work.
The calculated cost of the dryer is $475.
What is the cost of the dryer?Let's assume that the cost of the dryer is x dollars.
According to the problem, the cost of the washer is $54 more than the dryer.
Therefore, the cost of the washer is (x + $54).
We are given that the combined cost of the washer and the dryer is $1004. So we can set up the equation:
x + (x + $54) = $1004
Simplifying the equation:
2x + $54 = $1004
2x = $950
x = $475
Therefore, the cost of the dryer is $475.
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And 7/8 hours Greg reads 2/3 chapters. What is the unit rate in chapters per hour? Write your answer in simplest form
The unit rate for chapters per hour is calculated as 16/21
How to find the unit rateIn order to find the unit rate for the number of chapters Gregory reads in an hour, we must firstly divide the total amount of chapters read by the amount of hours devoted to the activity:
= 2/3 chapters ÷ 7/8 hours
= 2/3 chapters × 8/7 hours
= 16/21 chapters per hour
Consequently, the unit rate for chapters per hour is calculated as 16/21, which already exists in its most simplified form.
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The function h is given by h(x)=log_2(x^2 -6). For what positive value of x does h(x)=4?
The function h is given by h(x)=log2(x² -6). The positive value of x that makes h(x) equal to 4 is approximately 4.69
We have the function:
h(x) = log2(x² - 6)
We want to find the value of x that makes h(x) equal to 4:
h(x) = 4
log2(x² - 6) = 4
We can rewrite this equation as:
2⁴ = x² - 6
16 = x² - 6
x²= 22
x = √22 (because we are looking for a positive value of x)
Therefore, the positive value of x that makes h(x) equal to 4 is approximately 4.69 (rounded to two decimal places).
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