1.1 Factoring: x3-81
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0+b3 =
a3+b3
Check : 81 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
1.2 Find roots (zeroes) of : F(x) = x3-81
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is -81.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9 ,27 ,81
Let us test ....
Polynomial Roots Calculator found no rational roots
2.1 Solve : x3-81 = 0
Add 81 to both sides of the equation :
x3 = 81
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:
x = ∛ 81
Can ∛ 81 be simplified ?
Yes! The prime factorization of 81 is
3•3•3•3
To be able to remove something from under the radical, there have to be 3 instances of it (because we are taking a cube i.e. cube root).
∛ 81 = ∛ 3•3•3•3 =
3 • ∛ 3
The equation has one real solution
This solution is x = 3 • ∛3 = 4.3267
Answer:
The initial value is
✔ 1/3
The simplified base is
✔ 27
The domain is
✔ all real numbers
The range is
✔ y > 0
Step-by-step explanation:
Mark borrowed $5,500 at 11.5 percent for five years. What is his monthly payment?
A. $131.50
B. $173.50
C. $120.95
D. $150.22
Answer:
C option is correct
Step-by-step explanation:
Given: payment borrowed (P)= $5,500
rate (r)=11.5 or 0.115
Time(t)= 5 years
Compound monthly(k)=12
To find : Monthly payment (M)
Formula used : [tex]M= P/(\frac{1-(1+\frac{r}{k})^{-kn}}{\frac{r}{k}})[/tex]
Solution : [tex]M= 5500/(\frac{1-(1+\frac{0.115}{12})^{-60}}{\frac{0.115}{12}})[/tex]
by solving we get, M =$120.95
therefore, option C is correct
Find the difference between the product of 21.33 and 2.04 and the sum of 1.115, 3.18, 22.0613, and 12.2384. A. 5.1318 B. 38.5947 C. 4.9185 D. 43.5132
To find the difference, first find the product of 21.33 and 2.04, then find the sum of 1.115, 3.18, 22.0613, and 12.2384, and finally subtract the sum from the product. The correct answer is 4.9185.
To find the difference between the product of 21.33 and 2.04 and the sum of 1.115, 3.18, 22.0613, and 12.2384, follow these steps:
Calculate the product of 21.33 and 2.04:The difference is 4.9185, which corresponds to option C in the given choices.
98,464 rounded to the nearest ten thousand is
The base of a box is a rectangle. The width of the box is half its length. The height of the box is 0.5m. Find the volume of the box if the area of the base is 1.08m² less than the combined area of the sides.
Answer:
Step-by-step explanation:
We assume the box is open-top.
If the width of the box is represented by w, then its length is 2w. The area of the base is then w·2w = 2w².
The combined area of the sides is the product of the perimeter of the base, 2(w+2w) = 6w and the height, 0.5, so is 3w.
Since the base area is 1.08 m² less than the lateral area, we have
... 2w² = 3w -1.08
... 2w² -3w +1.08 = 0 . . . . rearrange to standard form
... w² -1.5w +0.54 = 0 . . . make the leading coefficient 1 (divide by 2)
... (w -0.9)(w -0.6) = 0 . . . factor
The width of the box may be either 0.6 meters or 0.9 meters.
The volume is the product of width, length, and height
V = w·2w·0.5 = w²
V = 0.36 m³ . . . or . . . 0.81 m³. . . . . . (there are two possible answers)
Define the function g(x) in terms of f(x)
Please answer with steps on how you solved it!
Jason bought a soft drink for $3 dollars and 9 candy bars. He spent a total of $30. How much did each candy bar cost?
A publisher claims that the average salary paid at its company is $37,500 but it could differ as much as $4,500. Write and solve an absolute value inequality to determine the range of salaries at this company.
|x − 37500| ≤ 4500; The salaries range from $33,000 to $42,000
|x − 37,500| ≤ 4500; The salaries are less than $33,000 or greater than $42,000.
|x − 37500| ≥ 4500; The salaries range from $33,000 to $42,000
|x − 37,500| ≥ 4500; The salaries are less than $33,000 or greater than $42,000.
Chance can wash the car in 2 hours, while Sandy can wash the same car in 4 hours. How many hours would it take for them to wash the car together?
A. 4/3
B. 3/4
C. 3
D. 4
Chance and Sandy together can finish the work in 4/3 hours.
What is the relation between time and efficiency?The relationship between time and efficiency is they are inversely proportional.
If one quantity increases other decreases and vice versa.
Given, Chance can wash the car in 2 hours.
∴ The efficiency of Chance is (1/2).
Sandy can wash the same car in 4 hours.
∴ The efficiency of Sandy is (1/4).
So, their combined efficiency is (1/2 + 1/4) = 3/4.
∴ Together they can finish the work in 4/3 hours.
learn more about time and efficiency here :
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Given the function f(x)=-5x^2-x+20 find f(3)
f(3) = -5*3^2 - 3 + 20 = -5*9 - 3 + 20 = -45 - 3 + 20 = -28;
Answer: - 28
Step-by-step explanation:
f(x) = -5x^2 - x + 20
To find f(3), we simply put x = 3 and then simplify
f(3) = - 5(3)^2 - 3 + 20
= -5(9) - 3 + 20
= -45 - 3 + 20
= -48 +20
= -28
f(3) = -28
Bill takes the commuter train to work every day. during the morning commute, a train arrives every 25 min. if bill arrives at the station at a random time for the morning commute, what is the probability that he will have to wait at least 5 min for a train?
HELP ME!!!!!!!!!!!!!!!1 translate each sentence into an equation
1- 7 berries are 5 less than twice the number of berries Mickey had for lunch.
2- Negative 4 times the difference of a number 7 is 12
The sum of two numbers is seventy-two. Twice one of them equals ten times the other. What are the numbers?
Point P is the incenter of triangle ABC, PZ = 7 units, and PA = 12 units.
The radius of the incircle centered at point P is ? units.
Answer:
The answer is 7 units
Step-by-step explanation:
I don't know but I got it right
I think you dont need to calculate anything they gave us the answer when they said PZ=7
Please help not really sure what to do here.
Divide 30 by 1/2 and add 10 what is the answer?
Of all rectangles with a perimeter of 21 21, which one has the maximum area? (give the dimensions.) let a be the area of the rectangle. what is the objective function in terms of the width of the rectangle, w?
To find the rectangle with the maximum area, we need to find the dimensions and write the objective function in terms of the width of the rectangle.
Explanation:To find the rectangle with the maximum area, we need to find the dimensions of the rectangle that will give us the maximum area. Let's assume the width of the rectangle is w. Since the perimeter of the rectangle is given as 21, we can write the equation:
2w + 2l = 21
We can rearrange this equation to solve for l:
2l = 21 - 2w
l = (21 - 2w)/2
The area of the rectangle can be calculated as:
A = w * l
Substituting the expression for l we obtained earlier:
A = w * (21 - 2w)/2
Expanding and simplifying this equation will give us the objective function in terms of the width w.
Triangle RST has vertices R(2, 0), S(4, 0), and T(–3, –1). The image of triangle RST after a rotation has vertices
R'(0, –2), S'(0, –4), and T'(–3, –1). Which rule describes the transformation?
R0, 90°
R0, 180°
R0, 270°
R0, 360°
Answer:
It should be C aka 270°
Step-by-step explanation:
Answer:
3 option is correct.
Step-by-step explanation:
Given Triangle RST has vertices R(2, 0), S(4, 0), and T(–3, –1). The image of triangle RST after a rotation has vertices R'(0, –2), S'(0, –4), and T'(–3, –1).
We have to tell the rule describes the transformation.
A rotation is transformation that turns a figure about a fixed point. An object and its rotation are the same shape, but the figures may turned or rotate in different directions.
Here, the coordinates shift after rotation according to the rule
(2,0) → (0,-2) i.e (x, y) → (y, -x)
As, Rotation of 270º on coordinate axes.
(x, y) → (y, -x)
which match to the given rule.
hence, 3 option is correct.
A computer company can sell 130 boxes of printer paper each week at a price of $9 per box. For each $1 increase in the price, it will sell 10 fewer boxes per week. What price per box will provide the maximum revenue for the computer company?
$9
$10
$11
$12
The maximum revenue for the computer company is achieved at the price of $11 per box which is calculated based on the decrease in quantity sold with every $1 increase in price.
Explanation:To find the price per box that will provide the maximum revenue for the computer company, we need to analyze how changes in price affect the number of boxes sold and the revenue generated. The company sells 130 boxes of printer paper each week at $9 per box. For each $1 increase in price, it will sell 10 fewer boxes.
Let's define the revenue function as R(p) where p is the price per box. The initial revenue is 130 boxes × $9. For each $1 increase, the revenue function becomes (130 - 10(p - 9)) × p. We need to find the value of p that maximizes this revenue function.
Let's calculate:
At $9: R($9) = 130 × $9 = $1170At $10: R($10) = (130 - 10(1)) × $10 = 120 × $10 = $1200At $11: R($11) = (130 - 10(2)) × $11 = 110 × $11 = $1210At $12: R($12) = (130 - 10(3)) × $12 = 100 × $12 = $1200We can see that the revenue is maximum at $11 per box. So, the price per box that will provide the maximum revenue for the computer company is $11.
What is the value of the expression “two less than six times the difference of a number and five” when n = 9?
Answer:
22
Step-by-step explanation:
The value of expression is 22.
Step-by-step explanation:
Given the statement “two less than six times the difference of a number and five” when n = 9. we have to find the value of expression formed by the above statement.
Difference of a number n and 5 gives (n-5)
Now, we have to write two less than 6 times the difference, we get the expression 6(n-5)-2
Here, the number n is 9
∴Value of expression is 6(9-5)-2=6(4)-2=24-2=22
Hence, the value of expression is 22.
A high-speed bullet train travel 900 feet in 5 seconds at this rate how far would it travel in 1 minute
Triangle LMN is positioned inside rectangle LKPN as shown in the diagram. Which is the area of △LMN△LMN?
Answer:
Step-by-step explanation:
The area of the triangle inside the rectangle is calculated by using the formula for the area of a triangle, 1/2 * base* height. Given the base is the length of the rectangle and the height is the breadth of the rectangle, we substitute these measurements into the formula to determine the area. Area of triangle LMN is 15.6 square cm.
Explanation:To determine the area of the triangle inside the rectangle, we first need to understand the relationship between these two shapes. Given the base of the triangle, LN, is the same as the length of the rectangle, 6.5 cm, and the triangle is between the parallel lines of the rectangle, we can infer that the height of the triangle is the same as the breadth of the rectangle, which is 4.8 cm.
Now, the area of a triangle is given by the formula 1/2 * base * height.
Substituting the given measurements into this formula, we get
= 1/2 * 6.5 cm (base) * 4.8 cm (height)
= 0.5 * 6.5 cm (base) * 4.8 cm (height)
= 15.6 square cm
This will give us the area of the triangle LMN.
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The complete question is given below:
Triangle LMN is positioned inside rectangle LKPN as shown in the diagram. Which is the area of △LMN?
The length of rectangle and triangle are same, 6.5 cm. The breadth of rectangle is 4.8 cm. The triangle lies between the parallel lines of rectangle with base LN.
If y varies directly as x, and y = 12 when x = 15, find x when y = 21.
Ezekiel was having a pizza party, so he bought 7 pizzas, each with 8 slices. Unfortunately, word of his party got out, and the party got out of control. Now, Ezekiel is trying to figure out how many people showed up. If he assumes that on average each person ate 3 1/2 slices, and if all the pizza was eaten, how many people would Ezekiel guess showed up?
What happens when you multiply a positive number by a positive number less than 1?
The length of a rectangle is the square root of 100 times an unknown number, x. The width is one-half an unknown number, y, less three-halves x. If the area of the rectangle is 125 cm2, find an equation that would give the width of the rectangle given the length.
HELP! CONFUSED ON WHAT TO DO!
Marathon Training
Amanda and Kevin are training for a marathon. To prepare for this marathon they have been training and tracking their progress periodically. In the first week of training Kevin ran an average of 9.00 minutes per mile. Later, in week five of training he ran an average of 6.50 minutes per mile. In the first week of training Amanda ran an average of 10.00 minutes per mile. Later, in week seven of training she ran an average of 5.00 minutes per mile. Assuming that Kevin and Amanda continue to train and improve their times at the same rate your task is to determine which week they will have the same average minutes per mile. We will assume that the relationship is linear as they will be training for a maximum of 10 weeks. To complete this task follow the steps below.
1. Determine the equation of a line in standard form that represents Kevin’s training progress. His progress corresponds to the points (1, 9.00) and (5, 6.50). 2. Determine the equation of a line in standard form that represents Amanda’s training progress. Her progress corresponds to the points (1, 10.00) and (7, 5.00) 3. Solve the system of equations (showing work would be awesome!)
4. In which week will Kevin and Amanda have the same average minutes per mile?
5. If Kevin and Amanda continue to train until week 16, what will their times be? 6. Do you believe a linear model best represents the relationship of the time of the runners and the weeks that passed?(Hint: look at question 5). What do you think this says about problems in the real world? Justify your thoughts in 3-4 sentences.
Sorry it's this long, but please, I really don't know where to start or what to use to solve! I really need the help to understand. If you can't answer all, it's OK, but at least 2 or more would be great, so that way, I'll know to where to start!
Delia uses 3.5skeins of yard to knit one scarf how many scarves can she completed if she has 19skeins of yarn
Which of the following shows the proper steps to construct a perpendicular bisector of a segment through any point?
Suppose we did an experiment to gather data on the number of un-popped kernels of microwave popcorn. Half of the students in your class took home a bag, put it in the freezer overnight, and popped it the next day. The other half took the bag, left it on their kitchen counter overnight, and popped it the next day. Each group counted the number of un-popped kernels. Are these data paired or independent
The data collected on the number of un-popped kernels of popcorn, where one group's bags were frozen and the other's were left at room temperature, are considered independent. This is because each bag was only subjected to one condition and was not measured in both conditions, which would be necessary for paired data.
In the question provided, students collected data on the number of un-popped kernels of microwave popcorn where one group froze the popcorn while the other left it at room temperature. To determine whether the data are paired or independent, we need to understand the nature of the experiment setup.
Since every student’s popcorn was subject to a unique condition (frozen or room temperature) and they're not measured in both conditions, we can deduce that the data here are independent. This means that the outcome of one does not influence the outcome of the other.
In comparison to a scenario where the same item or subject is tested under two different conditions the popcorn experiment does not meet this criterion; that would be an example of paired data. An unpaired test such as the unpaired t-test may be appropriate here if we want to assess the effects of freezing on the number of un-popped kernels compared to leaving them at room temperature. Each bag of popcorn represents an independent sample from the population of all popcorn bags.
What is the perimeter of a polygon with vertices at (-6,-1) (-3,-4) (6,5) and (3,8) ?