The factorization of[tex]8(p-q)^3-(p+q)^3 is 8(p-q)^3 - (p+q)^3 is [p - 3q][7p^2 - 8pq + 3q^2].[/tex]
The given expression is[tex]8(p-q)^{3}-(p-q)^{3}[/tex]
To factorize this expression, we can use the identity for the difference of cubes, which states that [tex]a^3 - b^3 = (a - b)(a^2 + ab + b^2).[/tex]
Let's apply this identity to the given expression:
[tex]8(p-q)^3 - (p+q)^3 = [2(p-q)]^3 - (p+q)^3.[/tex]
Now we can see that we have a difference of cubes:[tex](2(p-q))^3 - (p+q)^3.[/tex]
Using the identity, we can factorize this expression as follows:
[tex][(2(p-q)) - (p+q)][(2(p-q))^2 + (2(p-q))(p+q) + (p+q)^2].[/tex]
Simplifying further:
[tex][2p - 2q - p - q][(2(p-q))^2 + (2(p-q))(p+q) + (p+q)^2].[/tex]
Combining like terms:
[tex][p - 3q][(2(p-q))^2 + 2(p-q)(p+q) + (p+q)^2].[/tex]
Expanding the squared term:
[tex][p - 3q][4(p^2 - 2pq + q^2) + 2(p^2 - q^2) + (p^2 + 2pq + q^2)].[/tex]
Simplifying:
[tex][p - 3q][4p^2 - 8pq + 4q^2 + 2p^2 - 2q^2 + p^2 + 2pq + q^2].[/tex]
Combining like terms again:
[tex][p - 3q][7p^2 - 8pq + 3q^2].[/tex]
Therefore, the factorization of [tex]8(p-q)^3 - (p+q)^3 is [p - 3q][7p^2 - 8pq + 3q^2].[/tex]
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Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. You have a credit card with a balance of $754.43 at a 13.6% APR. You have $300.00 available each month to save or pay down your debts. a. How many months will it take to pay off the credit card if you only put half of the available money toward the credit card each month and make the payments at the beginning of the month? b. How many months will it take to pay off the credit card if you put all of the available money toward the credit card each month and make the payments at the beginning of the month? Be sure to include in your response: • the answer to the original question • the mathematical steps for solving the problem demonstrating mathematical reasoning
a. It will take 7 months to pay off the credit card.
b. it will take 4 months to pay off the credit card.
Since, APR stands for Annual Percentage Rate. It is the interest rate charged on a loan or credit card, expressed as a yearly percentage rate. The APR takes into account not only the interest rate, but also any fees or charges associated with the loan or credit card.
a. If you put half of the available money each month toward the credit card, then you are paying $150.00 per month towards the credit card balance.
We can use the formula for the present value of an annuity to find how many months it will take to pay off the credit card:
PV = PMT × ((1 - (1 + r)⁻ⁿ) / r)
where:
PV is the present value of the debt
PMT is the payment amount per period
r is the monthly interest rate
n is the number of periods
Substituting the values, we get:
754.43 = 150 × ((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)
Simplifying and solving for n, we get:
n = log(1 + (PV ×r / PMT)) / log(1 + r)
n = log(1 + (754.43×0.011333 / 150)) / log(1 + 0.011333)
n = 6.18
Therefore, it will take 7 months to pay off the credit card if you put half of the available money each month toward the credit card.
b. If you put all of the available money each month toward the credit card, then you are paying $300.00 per month towards the credit card balance.
754.43 = 300 ×((1 - (1 + 0.011333)⁻ⁿ) / 0.011333)
Simplifying and solving for n, we get:
n = log(1 + (PV × r / PMT)) / log(1 + r)
n = log(1 + (754.43× 0.011333 / 300)) / log(1 + 0.011333)
n = 3.43
Therefore, it will take 4 months to pay off the credit card if you put all of the available money each month toward the credit card.
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A line that passes through the points (–4, 10) and (–1, 5) can be represented by the equation y = - 5/3(x – 2). Which equations also represent this line? Select three options.
y=-5/3x-2
✅y=-5/3x+10/3
✅3y = –5x + 10
3x + 15y = 30
✅5x + 3y = 10
Can someone tell me if I chose the right answers
Options 2, 3, and 5 are correct representations of the line passing through the given points.
The equation y = -5/3(x - 2) represents a line passing through the points (-4, 10) and (-1, 5).
Let's verify each option:
y = -5/3x - 2: This equation does not represent the same line. The constant term is different (-2 instead of +10/3).
y = -5/3x + 10/3: This equation represents the same line. It has the same slope (-5/3) and the same y-intercept (10/3).
3y = -5x + 10: This equation represents the same line. It can be simplified by dividing both sides by 3, resulting in the same slope (-5/3) and the same y-intercept (10/3).
3x + 15y = 30: This equation does not represent the same line. The coefficients of x and y are different, resulting in a different slope.
5x + 3y = 10: This equation represents the same line. It has the same slope (-5/3) and the same y-intercept (10/3).
Therefore, options 2, 3, and 5 are correct representations of the line passing through the given points.
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The distance that a freefalling body falls in each second starting with the first second is given by the arithmetic progression 16, 48,80,112
find the distance, the body falls in the seventh second
Answer:
208 units
Step-by-step explanation:
The first term is given as 16, which means a = 16.
The second term can be obtained by adding the common difference to the first term: 16 + d = 48.
The third term is obtained by adding the common difference to the second term: 48 + d = 80.
The fourth term is obtained by adding the common difference to the third term: 80 + d = 112.
We can solve these equations to find the value of 'd':
16 + d = 48
d = 48 - 16
d = 32
48 + d = 80
32 + 48 = 80 (valid)
80 + d = 112
32 + 80 = 112 (valid)
Therefore, the common difference is 32.
Now that we have the common difference, we can find the distance the body falls in the seventh second.
The formula for finding the nth term of an arithmetic progression is:
a_n = a + (n - 1) * d
where a_n is the nth term, a is the first term, n is the position of the term, and d is the common difference.
Plugging in the values, we can find the seventh term:
a_7 = 16 + (7 - 1) * 32
a_7 = 16 + 6 * 32
a_7 = 16 + 192
a_7 = 208
Therefore, the distance the body falls in the seventh second is 208 units.
4. In a lab experiment, 5300 bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every 12 hours. Write a function showing the number of bacteria after t hours, where the hourly growth rate can be found from a constant in the function.
Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per hour, to the nearest hundredth of a percent.
The function representing the number of bacteria after t hours is N(t) = 5300 * (1 + 0.0592)^t, and the growth rate per hour is approximately 5.92%.
To represent the number of bacteria after t hours, we can use the exponential growth formula:
N(t) = N₀ * (1 + r)^t,
where N(t) is the number of bacteria after t hours, N₀ is the initial number of bacteria, r is the hourly growth rate, and t is the time in hours.
In this case, the initial number of bacteria is 5300, and the hourly growth rate can be determined from the doubling time of 12 hours. The growth rate can be calculated using the formula:
r = 2^(1/t_double) - 1,
where t_double is the doubling time in hours.
Substituting the given values, we have:
r = 2^(1/12) - 1 ≈ 0.0592.
Now we can write the function for the number of bacteria after t hours:
N(t) = 5300 * (1 + 0.0592)^t.
To determine the percentage of growth per hour, we can calculate the relative growth rate as a percentage:
percentage_growth = r * 100 ≈ 5.92%.
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Suppose that point P is the point on the unit circle obtained by rotating the initial ray through θ° counterclockwise. What is the length of segment OP?
The length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.
To determine the length of segment OP on the unit circle, we need to use trigonometry. Let's break down the problem step by step:
Definition: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system.
Initial Ray: The initial ray is a line segment that starts from the origin (0, 0) and extends to a point on the unit circle. It forms an angle with the positive x-axis.
Rotation: We are rotating the initial ray counterclockwise by θ degrees. This means we are essentially finding a new point on the unit circle based on the angle θ.
Trigonometric Functions: The trigonometric functions sine (sin) and cosine (cos) are particularly useful for calculating the coordinates of points on the unit circle.
sin(θ) gives the y-coordinate of a point on the unit circle.
cos(θ) gives the x-coordinate of a point on the unit circle.
Coordinates of Point P: Since we are rotating the initial ray counterclockwise by θ degrees, the coordinates of point P on the unit circle can be obtained as follows:
x-coordinate of P: cos(θ)
y-coordinate of P: sin(θ)
Distance from the Origin (Length of Segment OP):
Using the coordinates of point P, we can calculate the distance between the origin (0, 0) and point P using the distance formula.
The distance formula states that for two points (x1, y1) and (x2, y2), the distance between them is given by:
d = √((x2 - x1)² + (y2 - y1)²)
In this case, point P has coordinates (cos(θ), sin(θ)), and the origin is (0, 0). Thus, the distance (length of segment OP) is:
d = √((cos(θ) - 0)² + (sin(θ) - 0)²)
= √(cos²(θ) + sin²(θ))
= √(1) [Using the trigonometric identity: sin²(θ) + cos²(θ) = 1]
= 1
Therefore, the length of segment OP, which represents the distance from the origin to point P on the unit circle, is always equal to 1.
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Stephanie wanted to solve the equation 16=3x+1. Which inverse operations should she use to find the solution?
Subtraction and Division
Step-by-step explanation:Inverse operations help find the solution to equations.
Defining Inverse Operations
Firstly, let's define an operation. An operation in math is a function that can manipulate a value. Inverse operations are operations that are opposite operations that undo each other. For example, addition and subtraction are inverse operations because subtraction undoes addition. Multiplication and division are also inverse operations.
Solving the Equation
The equation 16 = 3x + 1 involves both addition and multiplication. So, to solve this, we can use the inverse operations of subtraction and division. First, subtract 1 from both sides.
15 = 3xThen, divide both sides by 3.
5 = xThis shows that by using subtraction and division, we can undo the addition and multiplication used in the equation. This allows us to find the value of x.
An online real estate website estimates that a fair price for Jerrold’s house would be $715,000. The market is
strong, so he is optimistic and puts the house on the market for $750,000. Two weeks later, the best offer
he’s gotten is $718,000, and so he accepts that offer. At what percent above the website’s estimate did he set
his asking price? At what percent below his asking price did he sell
Answer:
approximately 4.2667%
Step-by-step explanation:
To calculate the percentage above the website's estimate that Jerrold set his asking price, we can use the following formula:
Percentage above = ((Asking price - Website estimate) / Website estimate) * 100
Percentage above = (($750,000 - $715,000) / $715,000) * 100
Percentage above ≈ 4.895
Therefore, Jerrold set his asking price approximately 4.895% above the website's estimate.
To calculate the percentage below his asking price that Jerrold sold for, we can use the following formula:
Percentage below = ((Selling price - Asking price) / Asking price) * 100
Percentage below = (($718,000 - $750,000) / $750,000) * 100
Percentage below ≈ -4.2667
Therefore, Jerrold sold his house approximately 4.2667% below his asking price.
11 players are going to practice in the batting cage. how many different orders are possible
Answer:
Step-by-step explanation:
What is an improper fraction for 1 3/4
Answer:
An improper fraction for 1 3/4 is 7/4.
Step-by-step explanation:
[tex]\frac{7}{4}[/tex]
To find the improper fraction of a mixed number fraction. You first have to remove the whole number from the fraction (the big one bending the fraction)
Do this by multiplying the denominator (4) by the whole number (1)
4 x 1 = 4
Then add this number with the numerator (top number) which is 3.
4+3 = 7
Seven is our new numerator, our denominator stays the same (4)
So our new improper fraction is:
[tex]\frac{7}{4}[/tex]
Jim is participating in a 6-day cross-country biking challenge. He biked for 59, 52, 66, 45, and 68 miles on the first five days. How many
miles does he need to bike on the last day so that his average (mean) is 59 miles per day?
miles
Answer:
Jim needs to bike 64 miles on the last day to maintain an average of 59 miles per day.
Step-by-step explanation:
To find out how many miles Jim needs to bike on the last day to maintain an average of 59 miles per day, we can use the concept of averages.
The total distance Jim needs to bike over the 6 days to maintain an average of 59 miles per day can be calculated as follows:
Total distance = Average distance per day × Number of days
Total distance = 59 miles/day × 6 days = 354 miles
Jim has already biked a total of 59 + 52 + 66 + 45 + 68 = 290 miles over the first five days.
To find out how many miles Jim needs to bike on the last day, we subtract the distance he has already biked from the total distance needed:
Distance needed on the last day = Total distance - Distance already biked
Distance needed on the last day = 354 miles - 290 miles = 64 miles
Therefore, Jim needs to bike 64 miles on the last day to maintain an average of 59 miles per day over the 6-day cross-country biking challenge.
PLS HELP, BRAINLEST ANSWER GIVEN
This fraction is equivalent to
Answer: A -6x² + 2x -4
Step-by-step explanation:
[tex]\frac{-12x^{3}+ 4x^{2} -8x}{2x}[/tex] >Divide each of the top terms by 2x[tex]=\frac{-12x^{3}}{2x} +\frac{4x^{2} }{2x} -\frac{8x}{2x}[/tex]
= -6x² + 2x -4
Mathematicias pls help me ples
This problem is relatively simple. Before we begin, it might be beneficial to imagine the graph as a vertical number line. Imagining it this way makes it easy to count the units between them and gives you the answer: [tex]$\boxed{4}$[/tex].
Here's a slightly more advanced way to think about it:
The example question tells us to subtract the distance between the two points. The formula for this is [tex]\sqrt{\big{(}(x_{2}-x_{1})+(y_{2}-y_{1})\big{)}^2[/tex], but for this question, let's say that it is [tex]\sqrt{(y_{2}-y_{1})^2[/tex], or, to simplify it even more, [tex]$|y_{2}-y_{1}|$[/tex]. Now that we know the formula, we can substitute our y-values into the last formula and solve. Let's say that Point C is our first point and Point D is our second.
[tex]\big{|}(-8)-(-4)\big{|} = \big{|}-8+4\big{|} = \big{|}-4\big{|} = \boxed{4}[/tex] , so [tex]$4\text{ units}$[/tex] is our answer.
Disclaimer: Neither of the last two formulas I provided is the actual formula, just a version of the Distance Formula that might be easier to understand. The first formula is the actual thing, but you will encounter this in math later in life, probably around 8th or 9th grade.
what is equivalent to 3³
Answer:
Step-by-step explanation:
3x3x3= 27
Which scenarto could be modeled by the graph of the function A) & 10041.002)4
A
An ant colony that has an initial population ef 100 increases by 0.296 per year.
An ant colony that has an Infal population of 100 increases at a constant rate of 0.2 per year.
An ant colony that has an intal population of 100 decreases by 0.2% per year
D
An ant colony that has an Infial population of 100 decreases at a constant rate of 0.2 per year.
The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.
We have,
The function A(x) = 100 + 4x represents a linear relationship between the variable x (representing time in this case) and the variable A(x) (representing the population of the ant colony).
The term 100 in the function represents the initial population of the ant colony.
It indicates that at the starting point (x = 0), the population is 100.
The term 4x in the function represents the rate at which the population increases over time. Since the coefficient of x is positive (4), it indicates that the population is increasing.
For every unit increase in x (in this case, for every year that passes), the population increases by 4.
Therefore,
The function A(x) = 100 + 4x describes the scenario of an ant colony that starts with an initial population of 100 and experiences a constant rate of increase of 4 ants per year.
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URGENT!! ILL GIVE BRAINLIEST! AND 100 POINTS
PLEASE ANSWER FAST!!!
A) Two equations are,
S = 18 + 4t
L = 30 + 3t
Where, "t" represents the number of weeks that have passed.
B) It will take 12 weeks for Samuel and Lewis to have the same number of rocks in their collections.
C) when the amount of rocks in their collection is equal, Samuel and Lewis will have 66 rocks each.
Part A:
For a system of equations to represent the number of rocks in each person's collection.
Let "S" be the number of rocks in Samuel's collection and "L" be the number of rocks in Lewis's collection.
Hence, We can represent the situation as follows:
Equation 1:
S = 18 + 4t
Equation 2:
L = 30 + 3t
Where, "t" represents the number of weeks that have passed.
Part B:
To find out how many weeks it will take for Samuel and Lewis to have the same number of rocks in their collections, we need to set Equation 1 equal to Equation 2 and solve for "t".
This gives us the following:
S = L
18 + 4t = 30 + 3t
1t = 12
t = 12 weeks
Hence, It will take 12 weeks for Samuel and Lewis to have the same number of rocks in their collections.
Part C:
For number of rocks Samuel and Lewis will have when the amount of rocks in their collection is equal, we can substitute the value of "t" that we found in Part B into either Equation 1 or Equation 2.
Hence, substituting "t = 12" into Equation 1 gives,
S = 18 + 4(12)
S = 18 + 48
S = 66 rocks
Therefore, when the amount of rocks in their collection is equal, Samuel and Lewis will have 66 rocks each.
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is 72 and has an IRA with a fair market value of
Use
Table
45 to determine her required minimum distribution. b) What penalty would she incur if she failed to take the
distribution? c) What penalty would she have paid if she had made an early withdrawal of $10,000 to take a
vacation?
May Kawasaki's required minimum distribution (RMD) is $3,808. This is calculated by dividing her IRA balance of $98,000 by the distribution period of 26.2, which is found in the Uniform Life Table on page 45 for a 72-year-old.
How to explain the informationb) If May Kawasaki fails to take her RMD, she will incur a 50% penalty on the amount she should have withdrawn. In this case, the penalty would be $1,904.
c) If May Kawasaki had made an early withdrawal of $10,000 to take a vacation, she would have paid a 10% penalty on the amount withdrawn. In this case, the penalty would have been $1,000.
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1. May Kawasaki is 72 and has an IRA with a fair market value of $98,000. a) Use the Uniform Life Table on p. 45 to determine her required minimum distribution. b) What penalty would she incur if she failed to take the distribution? c) What penalty would she have paid if she had made an early withdrawal of $10,000 to take a vacation?
Solve the equation below for x by graphing
3x =8_2x
The solution to the equation 3x = 8 - 2x is x = 1.6.
To solve the equation 3x = 8 - 2x by graphing, we can plot the two sides of the equation as functions of x and find the point(s) where they intersect. Here's a step-by-step explanation:
Express the equation in the form of y = f(x). Rearrange the equation:
3x + 2x = 8
5x = 8
x = 8/5 or 1.6
Graph the functions y = 3x and y = 8 - 2x on the same coordinate plane. The line represented by y = 3x is upward sloping, and the line represented by y = 8 - 2x is downward sloping.
Plot the points (1.6, 3(1.6)) and (1.6, 8 - 2(1.6)) on the graph.
The point of intersection represents the solution to the equation. In this case, the lines intersect at (1.6, 4.8).
Therefore, the solution to the equation 3x = 8 - 2x is x = 1.6.
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Help pls i don't understand
ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.
What is Angle- Angle - Side theorem?The angle-angle-side theorem, or AAS, tells us that if two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.
If we consider triangle FSH and triangle FSI, we will observe the following;
angle HFS = angle IFSangle FSH = angle FSIlength HS = length SISo based on the angle-angle-side theorem, or AAS, we can see that triangle FSH is congruent to triangle FSI.
Thus, our answer will be; ΔFSH ≅ ΔFSI by the rule of angle-angle-side theorem, or AAS.
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If 15% of the customers total is $98,880, then the sum total equals what
Answer:
Step-by-step explanation:
∠RQT is a straight angle. What are m∠RQS and m∠TQS?
Answer:
m∠RQS = 102°
m∠TQS = 78°
Step-by-step explanation:
A straight angle is equal to 180 degrees. We will create an equation to solve for x.
9x° + 3° + 7x° + 1° = 180°
16x° + 4° = 180°
16x° = 176°
x = 11
Next, we will substitute this value into the expressions representing the angles.
m∠RQS = 9x° + 3° = 9(11)° + 3° = 102°
m∠TQS = 7x° + 1° = 7(11)° + 1° = 78°
A rectangular tree lot must have a perimeter of 100 uards and an area of at least 500 square yards. Describe the possible lengths of the tree lot.
The possible lengths of the tree lot are 25 + 5√5 yards and 25 - 5√5 yards.
Let's denote the length of the rectangular tree lot as "l" and the width as "w".
We know that the perimeter of a rectangle is given by the formula:
Perimeter = 2(l + w)
Given that the perimeter of the tree lot must be 100 yards, we can write the equation as:
2(l + w) = 100
Next, we know that the area of a rectangle is given by the formula:
Area = l × w
Given that the area of the tree lot must be at least 500 square yards, we can write the inequality as:
l × w ≥ 500
Now, let's solve the equations simultaneously to find the possible lengths of the tree lot.
Perimeter equation:
2(l + w) = 100
l + w = 50
w = 50 - l
Area inequality:
l × w ≥ 500
Substituting the value of w from the perimeter equation into the area inequality, we have:
l × (50 - l) ≥ 500
50l - l^2 ≥ 500
l^2 - 50l + 500 ≥ 0
Now, we need to find the values of l that satisfy the inequality. Since the coefficient of the squared term is positive, the graph of this quadratic opens upward. This means that the values of l that satisfy the inequality will be either the entire range of possible values or a portion of it.
To find the possible lengths, we can either factor the quadratic or use the quadratic formula. Let's use the quadratic formula:
l = (-(-50) ± √((-50)^2 - 4(1)(500))) / (2(1))
l = (50 ± √(2500 - 2000)) / 2
l = (50 ± √500) / 2
l = (50 ± 10√5) / 2
l = 25 ± 5√5
Therefore, the possible lengths of the tree lot are 25 + 5√5 yards and 25 - 5√5 yards.
In summary, the possible lengths of the rectangular tree lot are 25 + 5√5 yards and 25 - 5√5 yards, respectively.
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Help me! Make sure to do step by step! (I need to see the steps)
Simplify
(9x^8y^2z^6)^1/2
Answer: [tex]3x^4yz^3[/tex]
I'm hoping this is your equation: [tex](9x^{8}y^{2}z^{6})^{1/2}[/tex]
and not: [tex](9x^{8y^{2z^{z^6}}})[/tex]
Step-by-step explanation:
The square root of a number can be shown as a 1/2 power
We'll use the exponent rule:
= [tex]9^{1/2}(x^8 )^{1/2}(y^2)^{1/2}(z^6){1/2}[/tex]
Then we'll do each term individually
the square root of 9 is 3
for [tex](x^8)^{1/2}[/tex] the exponents multiply which give us [tex]x^4[/tex]
[tex](y^2)^{1/2}[/tex] gives us y
[tex](z^6)^{1/2}[/tex] gives us z^3
After doing all this we get [tex]3x^4yz^3[/tex]
What are the minimum and maximum values of the function?
The minimum value of [tex]f(x) = -2^3 \sqrt\(5-4x+3)[/tex] on the interval [1, 8] is -3 and the maximum value is 5. To find the minimum value, we can start by finding the critical points of the function.
The critical points are the points where the derivative of the function is equal to zero. In this case, the derivative of the function is
[tex]f'(x) = -2^3 \times (5-4x+3) ^(-3/2) \times (-4)[/tex]
The critical points of the function are x = 1 and x = 5.
We can now evaluate the function at each critical point and at the endpoints of the interval to find the minimum and maximum values. The values of the function at the critical points and at the endpoints are
x | f(x)
-- | --
1 | -3
5 | 5
8 | 9
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1/27^{4-x}=9^{2x-1}
Please if you could explain how you get your answer, that would be great.
The solution of the given equation is x = -10.
We are given that;
The equation 1/27^{4-x}=9^{2x-1}
Now,
This is an exponential equation that can be solved by using the properties of exponents and logarithms. Here are the steps to solve it:
Rewrite both sides of the equation using the same base. Since 27 and 9 are both powers of 3, we can use 3 as the base. We have:
(3(-3))(4-x) = (32)(2x-1)
Apply the power rule of exponents to simplify the expressions. The power rule states that (ab)c = a^(bc). We have:
3^(-12+3x) = 3^(4x-2)
Since the bases are equal, we can set the exponents equal to each other and solve for x. We have:
-12 + 3x = 4x - 2 -10 = x
Therefore, by the given equation the answer will be x = -10.
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Given f(x) = log2 (x+2), complete the table of values for the function -f(x) - 3. Show your work.
Answer:
-6
Step-by-step explanation:
If we convert the first function to the second form we get f(x) = -log2 (x+2) - 3. If we replace x with 6 we get ( -log2 8 ) -3. -log2 8 is equal to -3. -3 - 3 = -6.
4. The perimeter of the rectangle is represented by 8y metres and the area is represented by
(6y + 3) square metres.
X+8
x+6
a. Write two equations in terms of x and y: one for the perimeter and one for the area
of the rectangle.
b. Determine the perimeter and the area of the rectangle.
a) The two equations in terms of x and y: one for the perimeter and one for the area of the rectangle are:
y = 0.5x + 3.5
6y + 3 = x² + 14x + 48
b) The area and perimeter of the rectangle are:
Perimeter = 30 m
Area = 15 m²
How to find the perimeter and area of the rectangle?The formula to find the area of a rectangle is:
Area = Length * Width
The formula to find the perimeter of a rectangle is:
Perimeter = 2(Length + Width)
We are given that:
Perimeter = 8y meters
Area = (6y + 3) square meters
From the image, we see that:
Length = x + 6
Width = x + 8
Thus:
Perimeter equation is:
8y = 2(x + 6 + x + 8)
8y = 4x + 28
y = 0.5x + 3.5
Area equation is:
6y + 3 = (x + 6)(x + 8)
6y + 3 = x² + 14x + 48
Thus:
6(0.5x + 3.5) + 3 = x² + 14x + 48
3x + 24 = x² + 14x + 48
x² + 11x + 24 = 0
Using quadratic equation calculator gives:
x = -8 or -3
Thus, we will use x = -3 and we have:
Length = -3 + 6 = 3 m
Width = -3 + 8 = 5 m
Perimeter = 2(3 + 5) = 30 m
Area = 3 * 5 = 15 m²
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Type the correct answer in each box. Use numerals instead of words. If necessary, use / fc
The degree of the function (x) = (x + 1)2(2x-3)(x+2) is
Reset
, and its y-intercept
Next
Round to the nearest hundredth place.
7.2 ft
15.1 ft
The volume of the tarp shelter is 65.37 ft³ .
Given,
Conic tarp shelter with radius 5.1 ft and height 7.2 ft .
Now,
Volume of cone = 1/3 × π × r² × h
Substitute the values in the formula,
Volume of cone = 1/3 ×3.14 × (5.1)² × 7.2
Volume of cone = 65.37 ft³ .
Hence volume of tarp shelter will be 65.37 ft³ .
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In a health club, research shows that on average, patrons spend an average of 42.5 minutes
on the treadmill, with a standard deviation of 4.8 minutes. It is assumed that this is a normally
distributed variable. Find the probability that randomly selected individual would spent
between 30 and 40 minutes on the treadmill.
0,30
0.70
0.40
Less than 1%
Answer:
0.30
Step-by-step explanation:
To find the probability that a randomly selected individual would spend between 30 and 40 minutes on the treadmill, we need to calculate the z-scores corresponding to these values and then use the z-table or a statistical calculator to find the probability.
First, we calculate the z-scores using the formula:
z = (x - μ) / σ
where x is the value (in this case, 30 and 40), μ is the mean (42.5), and σ is the standard deviation (4.8).
For x = 30:
z = (30 - 42.5) / 4.8 ≈ -2.604
For x = 40:
z = (40 - 42.5) / 4.8 ≈ -0.521
Next, we look up the probabilities associated with these z-scores in the z-table or use a statistical calculator.
From the z-table or calculator, the probability corresponding to z = -2.604 is approximately 0.0047, and the probability corresponding to z = -0.521 is approximately 0.3015.
To find the probability between 30 and 40 minutes, we subtract the probability associated with z = -2.604 from the probability associated with z = -0.521:
P(30 ≤ x ≤ 40) = P(z = -0.521) - P(z = -2.604)
≈ 0.3015 - 0.0047
≈ 0.2968
Therefore, the probability that a randomly selected individual would spend between 30 and 40 minutes on the treadmill is approximately 0.2968, which is equivalent to 29.68%. Rounding up we will get 0.30.
Hope this helps!
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