Answer: A cubic polynomial function has the form f(x) = ax^3 + bx^2 + cx + d.
A horizontal tangent line is a line that is parallel to the x-axis and it occurs at a critical point of the function, where the derivative of the function is 0.
We know that a cubic function has 3 critical points and thus 3 derivative that could be 0, so to find the equation of the cubic function we will use these two points to find the value of a, b, c, and d.
We know that the critical points of a cubic function are found by solving f'(x) = 0, where f'(x) is the derivative of the function, so we have:
f'(x) = 3ax^2 + 2bx + c = 0
Now we can use the two points to find the value of a, b, c, and d.
We know that the point (1,-2) is on the graph of the function, so we can substitute it into the equation to find d:
f(1) = a + b + c + d = -2
And we know that the point (-1,2) is on the graph of the function, so we can substitute it into the equation to find d:
f(-1) = -a + b - c + d = 2
Now we can use these two equations to find the value of a, b, and c:
a + b + c + d = -2
-a + b - c + d = 2
2a + 2d = 0
a = -d
-a + b - c + d = 2
b - c = 4
3ax^2 + 2bx + c = 0
3(-d)x^2 + 2bx - c = 0
Now we can use these equations and the critical points to find the value of a, b, c, and d
3(-d)x^2 + 2bx - c = 0, (1,-2) => 3d+2b-c = 0
3(-d)x^2 + 2bx - c = 0, (-1,2) => 3d-2b+c = 0
b-c = 4
Solving these equations, we have:
a = -1/3
b = 1
c = -5/3
d = -2
And the equation for the function is:
f(x) = -1/3x^3 + x^2 - 5/3x - 2
We can now sketch the graph of the function by using the information that we have. The cubic function will have an vertex of symmetry at (0,-2) and two horizontal tangent lines at (1,-2) and (-1,2) .
Since a < 0, the function is concave down, so the graph will be opened downwards and will have a local minimum at (0,-2) .
It is important to note that this is one possible equation for a cubic function that has horizontal tangent lines at (1, −2) and (−1, 2), but there may be other possibilities.
Step-by-step explanation:
Find an equation of the sphere with center (2,-6,4) and radius 5.
Describe its intersection with each of the coordinate planes.
The equation of a sphere with center (a, b, c) and radius r is given by:
[tex](x - a)^2 + (y - b)^2 + (z - c)^2 = r^2[/tex]
Given the center (2,-6,4) and radius 5, we can substitute these values into the equation to find the equation of the sphere:
[tex](x - 2)^2 + (y + 6)^2 + (z - 4)^2 = 5^2[/tex]
So, the equation of the sphere is:
[tex](x-2)^2 + (y+6)^2 + (z-4)^2 = 25[/tex]
The sphere will intersect each of the coordinate planes in circles.
Intersection with the XY-plane:
The sphere intersects the XY-plane when z = 4.
Then, the equation of the circle on the XY-plane is [tex](x-2)^2 + (y+6)^2 = 5^2[/tex]
Intersection with the XZ-plane:
The sphere intersects the XZ-plane when y = -6.
Then, the equation of the circle on the XZ-plane is [tex](x-2)^2 + (z-4)^2 = 5^2[/tex]
Intersection with the YZ-plane:
The sphere intersects the YZ-plane when x = 2.
Then, the equation of the circle on the YZ-plane is [tex](y+6)^2 + (z-4)^2 = 5^2[/tex]
The sphere with center (2,-6,4) and radius 5 intersects each of the coordinate planes at circles.
These circles are defined by the equations [tex](x-2)^2 + (y+6)^2 = 25[/tex] on the XY-plane, [tex](x-2)^2 + (z-4)^2 = 25[/tex] on the XZ-plane, and [tex](y+6)^2 + (z-4)^2 = 25[/tex] on the YZ-plane.
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Oil is spilled onto a kitchen floor. The area covered by the oil at time t is given by the function A, where A(t) is measured in square centimeters and tis measured in seconds. Which of the following gives the rate at which the area covered by the oil is changing at time t=10? A A(10) B A(11) – A(9) с A(10) 10 A' (10)
The rate at which the area covered by the oil is changing at time t = 10 is A'(10).
Oil is spilled onto a kitchen floor. The function A, where A(t) is measured in square centimeters and t is measured in seconds, gives the area covered by the oil at time t.
Given that the region that was covered in oil at time t = A(t)
We must differentiate A(t) with respect to time t in order to calculate the rate of change in the area covered by the oil over time.
By differentiation:
[tex]\frac{dA}{dt}=A'(t)[/tex]
The rate at which the oily area is transforming is t = 10
[tex]\frac{dA}{dt}|_{t=10}=A'(t)|_{t=10}[/tex]
[tex]\frac{dA}{dt}=A'(10)[/tex]
Hence, the rate at which the area covered by the oil is changing at time t = 10 is A'(10).
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Solve pls this equation
Hey there!
If you are solving for x, then your final solutions are [tex]x=\sqrt{-y},-\sqrt{-y}[/tex].
Hope this helps!
A family has two cars. The first car has a fuel efficiency of 15 miles per gallon of gas and the second has a fuel efficiency of 30 miles per gallon of gas. During one particular week, the two cars went a combined total of 975 miles, for a total gas consumption of 45 gallons. How many gallons were consumed by each of the two cars that week?
Answer: We can start by using algebra to represent the information given in the problem. Let x be the number of gallons consumed by the first car and y be the number of gallons consumed by the second car.
We know that:
x + y = 45 (the total number of gallons consumed by both cars)
15x + 30y = 975 (the total number of miles driven by both cars, given the fuel efficiency of each car)
We have two equations and two unknowns, so we can use these equations to solve for the value of x and y.
From the first equation, we can solve for x:
x = 45 - y
We can substitute this expression of x into the second equation:
15(45 - y) + 30y = 975
Simplifying the left side of the equation:
675 - 15y + 30y = 975
Subtracting 675 from both sides:
15y = 300
And finally divide both sides by 15:
y = 20
So the second car consumed 20 gallons of gas that week.
We can substitute this value of y back into the equation x = 45 - y to find the value of x
x = 45 - 20 = 25
So the first car consumed 25 gallons of gas that week.
Step-by-step explanation:
Enter the equation for the graph y=[?] cos([?]x)
The equation for the graph will be
[tex]y=1 cos(3x)[/tex]
Define the function y = cos(2x + π/4) ?The function y = cos(2x + π/4) is a cosine function where the independent variable x is multiplied by 2 and then has π/4 added to it before the cosine is applied. The amplitude of this function is 1 and the period is π.
In this question
[tex]we assume, \\y=acos(bx)\\\\when x=0y=1\\\\y=acos(bx)\\\\1=acos0\\\\a=1\\so y=cos(bx)\\\\when x=\pi/3 y=-1\\y=cos(b.\pi/3)\\-1=cos1b.pi/3=cos(pi)\\b=3[/tex]
[tex]y=acosbx[/tex]
[tex]so, y=1.cos(3x)[/tex]
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the measure of the angle represented by the expression (7x+12)=1.?
The measure of the angle represented by the expression (7x+12) is 61°.
What is the measure of the angle represented by the expression (7x+12)?The measure of the angle represented by the expression (7x+12) is determined as follows:
The same-side interior angles are supplementary and their sum is 180°.
7x + 12 and 17x are same-side interior angles and are, therefore, supplementary.
7x + 12 + 17x = 180°
solving for x by summing similar terms:
24x + 12 = 180
24x = 168
Dividing both sides by 24
x = 7
Therefore,
7x + 12 = 7 * 7 + 12
7x + 12 = 61°
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Word Needed! *
The Bellah's pool is leaking. The pool leaked 3/4
gallon in 20 minutes, or 1/3hour. How many
gallons is the pool leaking each hour?
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The first diagram of the triangle DEF correctly represents the transformation.
What are transformations?Two-dimensional figures can be transformed mathematically in order to travel about a plane or coordinate system.
Dilation: The preimage is scaled up or down to create the image.
Reflection: The picture is a preimage that has been reversed.
Rotation: Around a given point, the preimage is rotated to create the final image.
Translation: The image is translated and moved a fixed amount from the preimage.
The preimage of the triangle is ABC and the image is DEF.
Therefore, ∠A = ∠D, ∠B = ∠E and ∠C = ∠F>
A 90° rotation counterclockwise will rotate the vertex C one-quarter left of a complete circle.
Therefore, The first triangle represents the correct transformation.
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Robin paid a $120 a month on a credit card. She was late on a payment so her monthly bill was raised to $150 a month. which of these shows how Robin can calculate the percent increase she was charged
Answer: One way to calculate the percent increase in Robin's monthly bill is to use the following formula:
% increase = (new amount - original amount) / original amount x 100
In this case, the original amount is $120 and the new amount is $150. So, the calculation would be:
% increase = (150 - 120) / 120 x 100 = 30 / 120 x 100 = 0.25 x 100 = 25
Therefore, the percent increase in Robin's monthly bill is 25%.
Another way to calculate the percent increase is to use the formula:
(New amount - original amount) / original amount
In this case: (150-120)/120 = 0.25, and multiply by 100 to get 25%
So, both formula gives the same answer 25%
Step-by-step explanation:
What is the linear velocity in miles per hour for a particle moving in a circular path at 200 revolutions per minute on a circle of radius 60 feet?
Answer: 1,200. None of the above.
Step-by-step explanation:
200 revolutions per minute times 60 feet. 1,200
Consider the following sample data.
14 9 19 18 5
8
Calculate the z-score for the following values.
a. 12
b. 6
c. 15
d. 13
a. The z-score for 12 is.
(Round to two decimal places as needed.)
The z-score for the given values is - 0280, -1.08, 0.49, 0.147 respectively.
What do z-scores mean?The z-score of a data point defines this precise distance in terms of standard deviations above or below the mean.
This formula is used to determine a z-score:
mean standard deviation for each data point
Means of the data points with a standard deviation of z
Below is the same equation in symbol form:
z= x−μ / σ
Here are some noteworthy z-score details:
When the z-score is high, the data point is.
A negative z-score indicates that a data point is below average.
If the z-score is close to 0.00, the data point is thought to be around average.
For the given data set,
14 9 19 18 5 8
Mean of the data =sum of observations / no. of observations = 73/6 = 12.16
Mean of the data is 12.16
Standard deviation = √∑(x - μ)² / n-1
= 5.70
a) The z- score of 12 is
z= x−μ / σ
z = 12 - 12.16 / 5.70
z = - 0280
b) x = 6,
z = 6 - 12.16/5.70
z = -1.08
c) x = 15
z = 15 - 12.16/5.70
z = 0.49
d) x = 13
z = 13 - 12.16 / 5.70
z = 0.147
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in the graph below, what best describes the average acceleration from 40 to 70 s?
The typical increase in speed from 40 to 70 seconds The magnitude is lower than the initial acceleration and is positive.
How do you describe average acceleration?The velocity change's average acceleration is referred to as this. To calculate any object's average acceleration, we divide the change in velocity by the amount of time that has passed. As a result, it is clear that the average acceleration is defined as the ratio of the change in velocity to the change in time over a certain interval.
The average acceleration is the velocity change divided by the time that has passed. A marble's average acceleration is 20 cm/s, for instance, if its velocity increases from 0 to 60 cm/s in three seconds. The marble's speed will increase by 20 cm/s every second as a result. The ball will therefore have an upwards-moving terminal velocity of 20 m/s after two seconds.
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The height in feet of a firework t seconds after it is launched is modeled by h(t) --16r2 + 105t + 10. Find its average speed from l to 3 seconds.
The change in height divided by the change in time yields the firework's average speed from t=1 to t=3 seconds.
Average change speed is equal to (h(3) – h(1)) (3 - 1)
h(3) = 16*32 + 105*3 + 10 = 536
h(1) = 16*12 + 105*1 + 10 = 126
Average speed = (536 - 126)/(3 - 1) = 410/2 = 205 ft/sec
h(t) = 16r2 + 105t + 10 models the height of a firework t seconds after it is released. We must determine the change in height divided by the change in time in order to get its average speed between t=1 and t=3 seconds.
By entering t = 3 into h(t), we can determine the height at 3 seconds, which gives us h(3) = 16*32 + 105*3 + 10 = 536. We can figure out the height.
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Part B: What is the probability that a randomly chosen person is a child? (3 points)
Part C: What is the probability that a randomly chosen person prefers running and is a child? (3 points)
Part D: What is the probability that a person prefers running given that the person is a child? (3 points)
Part E Are being a child and preferring running independent events? Explain. (5 points)
The probability that a randomly chosen person is a child, that a randomly chosen person prefers running and is a child is 12/25 , 1/ 6 respectively.
What is probability ?
Probability can be defined as the ratio of number of favourable outcomes and number of total outcomes.
Given in the table ,
Probability that a randomly chosen person is a child
= 48 / 100
= 12/25
Probability that a randomly chosen person prefers running and is a child
= 6/48
= 1/8
Probability that a person prefers running given that the person is a child
= 6/48
=1/8
Therefore, The probability that a randomly chosen person is a child, that a randomly chosen person prefers running and is a child is 12/25 , 1/ 6 respectively.
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Which quadrilaterals have diagonal that bisect their angles?
Rhombus and Square are the quadrilaterals where diagonals are equal and bisect each other at 90°.
What is quadrilateral?A quadrilateral is a four-sided closed shape, where sum of the interior angles equals 180°.
Here,
For the quadrilateral there are only two quadrilaterals where diagonals bisect each other at 90° and will be of equal length of measure are square and rhombus.
Thus, Rhombus and Square are the quadrilaterals where diagonals are equal and bisect each other at 90°.
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Two trains leave towns 476 kilometers apart at the same time and travel toward each other. One train travels 22 km/h faster than the other. If they meet in 2 hours, what is the rate of each train?
Answer:
Train 1: 130 km/h
Train 2: 108 km/h
Step-by-step explanation:
Let train 1 be the faster train.
Train 1:
distance = d
speed = s
time = 2 hours
Train 2:
distance = 476 - d
speed = s - 22
time = 2 hours
speed = distance/time
distance = speed × time
Train 1:
d = 2s
Train 2:
476 - d = 2(s - 22)
d = 2s
476 - d = 2s - 44
d = 2s
476 - 2s = 2s - 44
520 = 4s
s = 130
s - 22 = 108
Train 1: 130 km/h
Train 2: 108 km/h
the pressure gauge on a cylinder of gas registers the gauge pressure, which is the difference between the interior pressure and the exterior pressure p0. let's call the gauge pressure pg. when the cylinder is full, the mass of the gas in it is mi at a gauge pressure of pgi. assuming the temperature of the cylinder remains constant, show that the mass of the gas remaining in the cylinder when the pressure reading is pgf is given by mf
The gauge pressure, or the difference between the inner pressure and the external pressure p0, is what the pressure gauge on a gas cylinder measures. Give the gauge pressure the letter pg.
The gas in the cylinder weighs mi at pgi gauge pressure when it is full. We can apply the ideal gas formula, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature, assuming the cylinder's temperature stays constant.
We can assume that the product nRT is constant because the temperature is constant. Consequently, we can write:
(pgf + p0) = (pgi + p0)(Vi) (Vf)
where Vi is the cylinder's initial volume and Vf is the cylinder's final volume when the pressure is pgf.
We also know that the gas's mass, m, is determined by multiplying the number of moles, n, by the molar mass, M, or m = nM.
The ideal gas law can be used to express the number of moles in terms of volume and pressure:
m/M = P(V/RT) = n
Thus, mf = pgf(Vf/RT).
M equals (pgf + p0)(Vf/RT).
(Vi/RT) Mi (pgi + p0)
where mf is the mass of gas still present in the cylinder at pgf pressure.
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Suppose a company has fixed costs of $40,800 and variable cost per unit of 1/3 x + 222 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is 1946 − 2/3 x dollars per unit.
The break-even points at which the sum of the fixed and variable costs (40,800 + (1/3)·x + 222) is equivalent to the revenue (1946·x - (2/3)·x²) are; x = 21, 2897
What is a break-even point?The break-even point in cost accounting is the point where the total cost of production is equivalent to the total revenue from the sale of the products.
The fixed cost of the company = $40,800
The variable cost = (1/3)·x + 222 Dollars
x = The total number of units sold
The selling price of the product = 1946 - (2/3)·x Dollars
A possible question, obtained from a similar question is the value of the number of units sold at break even point
The break even point where the cost and revenue are equivalent, can be found as follows;
Revenue = Selling price × Number of units sold, therefore;
Revenue = (1946 - (2/3)·x) × x = 1946·x - (2/3)·x²
At the break even point, we get;
Sum total of the cost = Revenue
Therefore;
Fixed cost + Variable cost = Revenue
40,800 + (1/3)·x + 222 = 1946·x - (2/3)·x²
1946·x - (2/3)·x² - (40,800 + (1/3)·x + 222) = 0
(-2·x² + 5837·x - 123066)/3 = 0
x = (-5837 ± √(5837² - 4×(-2)×(-123066)))/(2×(-2))
x = (-5837 ± √(33086041))/(-4)
x ≈ 21 or x ≈ 2897
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A fashion store is planning its end–of–season sale. It sells 40 of its signature jeans every week at $175 per pair. Last year, the store's sales increased to 60 pairs per week when the price was dropped by $25. By what amount should the store discount the price this year to maximize its revenue?
A. $112.50
B. $103.50
C. $62.50
D. $71.50
Answer:
A
Step-by-step explanation:
Every $1 decrease in price corresponds to a 0.8 increase in the number of pairs of jeans sold.
So, if [tex]x[/tex] pairs of jeans are sold, the revenue in dollars is [tex](175-x)(40+0.8x)[/tex].
The roots of this function are [tex]x=175[/tex] and [tex]x=-50[/tex]. The maximum is achieved halfway between these roots at [tex]x=62.5[/tex].
When 62.5 pairs of jeans are sold, the price is $112.50.
Based on the information, the store should discount $112.50 from the price to maximize its revenue.
How to calculate the amount of discount that the store should make?To calculate the amount of discount that the store must make to maximize its profits we must perform the following mathematical operation:
First we have to analyze the available data which are:
40 exclusive jeans/week ($175 each)60 exclusive jeans/week ($150 each)According to the above information, the problem could be answered with the following mathematical formula:
(175 - x)(40 + 0.8x)
x = 175
x = -50
x = 62.50.
According to the above, the correct answer is $112.50 (option A).
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When u and v are nonzero vectors, Spanlu,v) contains only the line through u and the line through v and the origin.
A. False. Span(u,v) includes linear combinations of both u and v. B. False. Span(u,v) will not contain the origin. C. True. Span(u,v) is the set of all scalar multiples of u and all scalar multiples of v
Span(u,v) is the set of all linear combinations of u and v, including scalar multiples of both. It does not include the origin.
Span(u,v) is the set of all linear combinations of u and v, meaning any combination of u and v multiplied by scalars. This includes scalar multiples of both u and v, but does not include the origin. To calculate Span(u,v), first write u and v as vectors u = (u1,u2,...,un) and v = (v1,v2,...,vn). Then, any linear combination of u and v can be written as c1u + c2v, where c1 and c2 are scalars, and the set of all such linear combinations is Span(u,v). For example, if u = (1,2) and v = (3,4), then Span(u,v) = {(1,2), (3,4), (4,6), (5,8), (7,10), ...}. In other words, Span(u,v) is the set of all scalar multiples of u and all scalar multiples of v.
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[tex]\frac{csc x -1}{cot x} = \frac{cot x}{csc x + 1}[/tex]
Answer:
The equation [tex]\frac{csc x -1}{cot x} = \frac{cot x}{csc x + 1}[/tex] can be simplified by cross-multiplying the fractions and then solving for x. This will result in the equation [tex]csc^2 x - 1 = 0[/tex], which can be simplified to [tex]csc x = 1[/tex], and thus [tex]x = \frac{\pi}{2} + n\pi[/tex], where n is any integer.
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a number n is said to be squarefree is the largest square that divides n is 1. show that any square-free integer can be uniquely written as the product of a squarefree integer and a square
Any square-free integer can be expressed as the product of a square-free integer (which has no square factors) and a perfect square (which has only square factors). This product is unique for any given square-free integer.
Any square-free integer can be expressed as the product of a square-free integer and a perfect square. This is because a square-free integer has no square factors, so the product of a square-free integer and a perfect square would have only square factors, and thus be square-free. To show that this product is unique for any given square-free integer, we can use the fact that any integer can be expressed as the product of its prime factors and that a square-free integer has no square factors. This means the prime factors of the square-free integer are all distinct, and thus the product of the square-free integer and the perfect square can be expressed as the product of the prime factors of the square-free integer and the prime factors of the perfect square. Since the prime factors of the perfect square are all the same, the product is uniquely determined by the prime factors of the square-free integer. Thus, the product of a square-free integer and a perfect square is unique for any given square-free integer.
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Identify the slope and y intercept of each graph. Drag the 3 points to the correct places on the graph drag and adjust the line so it goes through the points on the line
Y = 2x - 5
M =
B =
Answer:
slope: 2 , y-int: (0, -5)
Step-by-step explanation:
Choose the blanks
The linear relationship is or is not
And the graph passes or does not pass
Because the ratios between each set of numbers are equal, and the graph passes through the point, the linear relationship is proportionate (0,0).
What does the word "proportional" mean?Proportional connections are those in which the ratios of the two variables are equal. Another way to think about them is that one variable is always a constant value multiplied by the other in a proportional relationship. The "constant of proportionality" is the name given to this variable.
The origin, points (2,14.5), (4,29), and (6,43.5) are provided (0,0).
We are aware that the ratios are equivalent in a proportional connection, i.e.
So,
⇒ 14.5/2 = 29/4 = 43.5/6 = 7.25
The linear relationship is a proportionate relationship since the ratios are equal.
The given situation's graph has been drawn, and it is evident that the graph passes through its origin, or "zero," which is (0,0)
Since the ratios between each set of data are equal, the linear relationship can only be a proportional relationship because the graph passes through the point (0,0).
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which model can be used to find 4/5 1/10
Which ordered pair is a solution to the linear system?
2x + y = 5
x – y = 13
Which of the following are properties of a kite? Choose ALL that apply.
Diagonals are perpendicular
Two pairs of consecutive sides are congruent
1 diagonal bisects the other
Opposite angles are congruent
Answer:
Diagonals are perpendicular
1 diagonal bisects the other
Opposite angles are congruent
To friends shop for fresh fruit. Jaxson buys a watermelon for $8.25 and 3 pounds of cherries. Tim buys a pineapple for $3.15 and 2 pounds of cherries. Use the variable P to represent the price in dollars per pound of cherries. write and simplify an expression to represent how much more Jaxson spent.
If two friends shop for fresh fruit. Jaxson buys a watermelon for $8.25 and 3 pounds of cherries. The expression to represent how much more Jaxson spent is :5.1 +p.
How to find the expression?P = price of the cherries per pound
Price of cherries bought by Jaxson= $3p
Total amount spent by Jaxson = $8.25 + $ 3p
Price of cherries bought by Tim = $2p
Total amount spent by Tim = $3.15 + $ 2p
Now let find how much more Jaxson spent.
8.25 + 3p - 3.15 + 2p
5.1 + 3p - 2p
Combine like terms
5.1 +1p
Multiply by 1
5.1 + p
Therefore the expression to represent how much more Jaxson spent is :5.1 +p.
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Suppose P dollars is invested with an annual interest rate of 3.5% compounded semiannually. Use the laws of exponents to find the equivalent annual rate.
Answer:
3.53%
Step-by-step explanation:
annual 3.5%
So half a year =1.75% and compound twice a year.
equivalent annual rate is (1+1.75%)^2 -1 =.0353 =3.53%
Rewrite 8.24 x 10 6 in standard form
A 824,000
B 8,240,000
C 8,240,000,000
D 82,400,000
Answer:
answer is B) 8 240 000
(: have a nice day
Answer:
B 8,240,000
Step-by-step explanation:
your welcome