There are no values within the interval that satisfies the condition stated by Rolle's theorem.
What numbers does satisfy Rolle's theorem?In accordance to Rolle's theorem, a function existing within an closed interval [a, b] has at a least a value c within that interval such that the first derivative f'(c) is equal to the slope of the secant line that passes through the two interval limits. That is to say, Rolle's theorem is defined by the following formula:
f'(c) = [f(b) - f(a)] / (b - a), where a < c < b
Where:
f(a), f(b) - Values of the function evaluated at each interval limit.f'(c) - First derivative of the function evaluated at x = c.First, evaluate the function f(x) at x = - 4:
x = - 4
f(- 4) = (- 4)³ - 4 · (- 4)² - 16 · (- 4) + 8
f(- 4) = - 64 - 64 + 64 + 8
f(- 4) = - 56
Second, evaluate the function f(x) at x = 4:
x = 4
f(4) = 4³ - 4 · 4² - 16 · 4 + 8
f(4) = 64 - 64 - 64 + 8
f(4) = - 56
Third, determine the first derivative of the function f(x) and evaluate it at x = c:
f(x) = x³ - 4 · x² - 16 · x + 8
f'(x) = 3 · x² - 8 · x - 16
x = c:
f'(c) = 3 · c² - 8 · c + 16
Fourth, find the roots of the quadratic function found in the previous step.
3 · c² - 8 · c + 16 = 0
In accordance with the quadratic formula, the roots of the expression are 4 / 3 + i 4√2 / 3 and 4 / 3 - i 4√2 / 3, which are not real numbers.
Fifth, determine whether Rolle's theorem are applicable on the roots found in the previous step.
Since the roots of the polynomial 3 · c² - 8 · c + 16 are not real numbers, there are no number within the interval that satisfies Rolle's theorem.
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a flagpole has a height of 15 yards it will be supported by 3 cables edach of which is attached to the flagpole at a point 3 yards below the top of the pole and attached to the ground at a point that is 9 yards from the base of the pole find the total number of yards of cable that will be required
There are 45 yards of cable that will be required.
Let us find the total number of yards of cable that will be required.Since we have given that
Height of cable = 15 yards - 3 yards = 12 yards
Distance from pole = 9 yards
cable is the hypotenuse of the right-angled triangle.
So, it becomes,
[tex]c^{2} = b^{2}+p^{2}[/tex]
[tex]c^{2} = 12^{2}+9^{2}[/tex]
[tex]c^{2} = 144+81[/tex]
[tex]c^{2} = 225[/tex]
[tex]c = \sqrt{225}[/tex]
c=15 yards
Since the number of cables supported = 3
So, the number of yards of cable that will be required would be
= 3 x 15 = 45
Hence, there are 45 yards of cable will be required.
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which polynomial function has a leading coefficient of 1 and roots (7+i) and (5-i) with multiplicity 1?
The polynomial function which has a leading coefficient 1 is (x-(7+i))(x-(5-i))(x-(7-i))(x-(5+i))
Given that a polynomial function has leading coefficient 1 and roots (7+i) and (5-i) with multiplicity 1.
We want to find the polynomial function.
As we know that the complex conjugate properties of the roots of a polynomial states that if a+bi is a root of a polynomial then their complex conjugate a-bi is also a root.
So, according to this property (7-i) and (5+i) with multiplicity 1 is also a root of polynomial.
Thus, all the roots are x=7+i, x=7-i, x=5+i and x=5-i
So, the polynomial is f(x)=(x-(7+i))(x-(5-i))(x-(7-i))(x-(5+i))
Hence, the polynomial function which has a leading coefficient 1 and roots (7+i) and (5-i) is (x-(7+i))(x-(5-i))(x-(7-i))(x-(5+i)).
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Heeelllllllllllp Simplify the expression completely.
√144 +18 √12-5√64
Answer:
A
Step-by-step explanation:
using the rule of radicals
[tex]\sqrt{a}[/tex] × [tex]\sqrt{b}[/tex] ⇔ [tex]\sqrt{ab}[/tex] , then
[tex]\sqrt{12}[/tex]
= [tex]\sqrt{4(3)}[/tex]
= [tex]\sqrt{4}[/tex] × [tex]\sqrt{3}[/tex]
= 2[tex]\sqrt{3}[/tex]
Then
[tex]\sqrt{144}[/tex] + 18[tex]\sqrt{12}[/tex] - 5[tex]\sqrt{64}[/tex]
= 12 + 18(2[tex]\sqrt{3}[/tex] ) - 5(8)
= 12 + 36[tex]\sqrt{3}[/tex] - 40
= - 28 + 36[tex]\sqrt{3}[/tex]
Four different linear functions are shown below. Drag the representation of each function in order from greatest y intercept ti least y intercept
Order of greatest y intercept to least y intercept,
1). A linear function has been given as,
y = [tex]\frac{1}{2}[/tex]x + 4
Y intercept = 4
2). The graph,
Y intercept = (2)
3). Graph represented by table
Y intercept = 1
4). A line passes through two points (-2, -3) and (4, 0)
Y intercept = -2
What is Y- intercept?
It is the total distance from the origin and the intersection of y axis and the formed line. It may be positive, negative or zero.
From the graph,
y-intercept = (2)
A line passes through two points (-2, -3) and (4, 0),
Slope of the line = [tex]\frac{y2 - y1}{x2 - x1}[/tex]
= [tex]\frac{0-(-3)}{4-(-2)}[/tex]
= 1/2
Equation of a line passing through a point (x', y') and slope = m
y - y' = m(x - x')
If the point is (4, 0) and slope = 1/2
y - 0 = 1/2 ( x - 4 )
y = 1/2x - 2
y-intercept of this line = (-2)
A linear function has been given as,
y = [tex]\frac{1}{2}[/tex]x + 4
Comparing it with the slope intercept equation,
y = mx + b
b = y-intercept = 4
From the given table,
For x = 0, f(x) = 1
Therefore, y-intercept = 1
Order of greatest y intercept to least y intercept,
1). A linear function has been given as,
y = [tex]\frac{1}{2}[/tex]x + 4
Y intercept = 4
2). The graph,
Y intercept = (2)
3). Graph represented by table
Y intercept = 1
4). A line passes through two points (-2, -3) and (4, 0)
Y intercept = -2
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Nigel is selling candy bars for his outdoor club. He is averaging 4 bars every 3 days. If he continues at this rate, how many days will it take to sell his box of 35? Answer in whole number form and round up to the next whole number.
The number of days that it will take to sell his box of 35 is 21 days.
What is a ratio?Ratio simply demonstrates how many times one number can fit into another number. Ratios contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).
In this case, Nigel is selling candy bars for his outdoor club and he is averaging 4 bars every 3 days.
Let the number of days be x.
This will be:
3/x = 4/35
Cross multiply
4x = 3 × 35
x = (3 × 35)/4
x = 105 / 4
x = 21
It will take 21 days
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79 ala decena mas cercana
Answer:
80
Step-by-step explanation:
Please help me solve this
The value of a=2 and b= 4.
What is Equation?Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given:
a + 1 / b-1 = 5/1 and a-1/ b+ 1 = 1/1
So, a + 1 / b-1 = 5/1
now, cross multiplying
a+ 1 = 5b - 5
a- 5b = -6............(1)
Also, a-1/ b+ 1 = 1/1
a-1 = b + 1
a - b = 2 ..............(2)
solving Equation (1) and (2) , we get
2+ b - 5b= -6
-4b = -8
b= 2
and, a = 2 + 2 = 4
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The half-life of a radioactive kind of americium is 7,380 years. How much will be left after
22,140 years, if you start with 188,320 grams of it?
grams
The height, in feet, of a flying disc after t seconds is modeled by the equation h(t)=−4.5t^2+19.6t+3. What does the 3 in the equation represent?
The 3 in the equation is the starting point
Function given:
h(t) = −4.5t2 + 19.6t + 3
At t =0 we get h(0) = 3. This is called the y-intercept
It is the start point indicating the disc was thrown at 3 feet of height.
Using the common convention in geometry the horizontal axis in a graph in represented by the x-axis and the vertical axis in a graph is represented by the y-axis. The intercepts are the points on the graph at which the graph crosses the two axes. The point where the graph is drawn crosses the horizontal axis is called the x-intercept or the x-axis, and when the graph crosses the vertical axis is called as the y-intercept or the y-axis.
I hope this answer helps.
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The dimensions of a rectangular garden are 15 yards by 20 yards. A sidewalk, one yard wide, is to be built around the outside of the garden. What will be the area of the sidewalk?
Answer:
74 square yards
Step-by-step explanation:
Area of two-dimensional shapes:Area of rectangular garden:
l = 20 yds
w = 15 yds
Area of the rectangular garden = l *w
= 20 * 15
= 300 square yards.
Area of the outer rectangle (rectangular garden and the sidewalk):
length = 20 + 1 + 1 = 22 yds
width = 15 + 1 + 1 = 17 yds
Area of outer rectangle = 22 * 17
= 374 square yards
To find the area of the sidewalk, subtract the area of rectangular garden from the outer rectangle.
Area of the sidewalk = area of outer rectangle - area of sidewalk
= 374 - 300
= 74 square yards
the tiny country of fremont has a population of 100,000 people. in 2009, there were 2,000 births, 500 deaths, 200 emigrants, and 100 immigrants. what is the population growth rate (r) for 2009?
The population growth rate (r) for 2009 in the tiny country of fremont is found as 1.4.
Describe the term population growth rate (r)?The population growth rate (r) shows how fast a population ages over time. A population is growing if the growth rate is positive. A negative growth rate signals a declining trend. A birth rate (b) and mortality rate are the two primary elements that influence population increase (d). People leaving the population to move to another territory or immigrating from a different region may also have a consequence on population increase (emigration, e).All these elements are taken into account when formulating the population growth formula.
r = [(b + i) - (d + e)]/1000
r = population growth rateb = birth rate; 2,000i = immigration rate; 100d = death rate; 500e = emigration rate; 200Put the values in the formula.
r = [(2,000 + 100) - (500 + 200)]/1000
r = 2100 - 700/ 1000
r = 1.4
Thus, the population growth rate (r) for 2009 in the tiny country of fremont is found as 1.4.
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help meeeeeeeeeeeeeeeeeee pleaseeee rn rnnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
help meeeeeeeeeeeeeeeeeee pleaseeee rn rnnnnnnnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
The equation of the parabola that similar to f(x) = [tex]7x^2[/tex] but the vertex is (3, 5) in the standard form is y = [tex]7(x-3)^2[/tex] + 5
The equation of the parabola
f(x) = [tex]7x^2[/tex]
The standard vertex form of a parabola is
y = [tex]a(x-h)^2[/tex] + k
The coordinates of the vertex of a parabola = (3, 5)
Where (h, k) is the coordinates
From the given function of the parabola f(x) = [tex]7x^2[/tex]
The value of a = 7
Substitute the values in the vertex form of a parabola
y = [tex]7(x-3)^2[/tex] + 5
Hence, the equation of the parabola that is similar to f(x) = [tex]7x^2[/tex] but the vertex is (3, 5) in the standard form is y = [tex]7(x-3)^2[/tex] + 5
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The mean, median and mode of the numbers below are the same,
what is the value of a?
6, 2, a, 7, 4, 9,6
Answer:
8
Step-by-step explanation:
Put the numbers in order from lowest to highest. Add all the numbers together. We will ignore a for this step. You should get 34 when adding it all together. Then if you were to add 8 to that 34 you'd get 42. Divide 42 by the amount of numbers you have which is 7. Your answer is 6. Now to make sure 8 is our value. We want to make sure the median is also 6. to find the median you need to have the numbers in order for lowest to highest and find the number in the middle. Don't forget to add your new number 8 into that order. You should get 6 as the median. Therefore our answer is 8
-13x + x = 8x – 20x help me pls
The solution of the equation given as -13x + x = 8x - 20x is infinite many solutions
How to evaluate the equation?From the question, the equation is given as
-13x + x = 8x - 20x
To start with, we need to rewrite the expression
So, we have the following representation
x - 13x = 8x - 20x
Solving further, we need to evaluate the like terms
So, we have the following equation
-12x = 8x - 20x
Evaluate the like terms again
So, the equation becomes
-12x = -12x
Both sides of the above equation are the same
i.e. -12x is on both sides of the equation
When we have a solution that all sides of the equation have the same value, then the solution is that the equation has infinite many solutions
Hence, -13x + x = 8x - 20x has infinite solutions
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The ordered pairs show the number of hours a student studied in a week, x, and that student’s exam score, y.
(1,70) (3,75) (5,80) (9,95) (12,98)
The equation y=2.7x+67 can be used to model the relationship between the number of hours a student studied in a week and their exam scores. Choose a number from each box to make the statement true.
The y-intercept shows that a student who studies____________
A: 0
B: 2.7
C: 67
D: 69.7
hours can expect to get a score of _______on the exam.
A: 0
B: 2.7
C: 67
D: 69.7
Answers:
1st box: 0
2nd box: 67
===============================================
Explanation:
The y intercept always occurs when x = 0. This is when the function curve either crosses or touches the vertical y axis. Therefore, the y intercept is when the student studies 0 hours
Plug x = 0 into the equation
y = 2.7x + 67
y = 2.7*0 + 67
y = 67
Or you can note that the equation is of the form y = mx+b
m = slope = 2.7b = 67 = y interceptThe y-intercept of 67 tells us that the person studying for 0 hours is estimated to get a score of 67
two equal rectangular lots are enclosed by fencing the perimeter of a rectangular lot and then putting a fence across its middle. if each lot is to contain 300 square feet, what is the minimum amount of fence (in ft) needed to enclose the lots (include the fence across the middle)?
The minimum amount of fence needed to enclose the lots is 120ft.
Adding areas of both fields, the combined area comes about to be 600sq ft.
The dimensions of the new plot are x length and y breadth. According to the question, there is a fence in the middle with y breadth.
g(x, y, z) = xy - 600
g(x, y, z) = 2x + 3y
∇g = ∇ (xy - 600) = (y, x)
∇f = ∇(2x + 3y) = (2, 3)
Now from the Lagrange method,
∇f = λ∇g
2 = λy
3 = λx
Now to get the value of lambda,
(2/y) = (3/x) = λ
x = 1.5y
Solving now we get,
1.5y² = 600
y = 20
x = 1.5 × 20 = 30
Thus after getting (x,y) we can put the value in the function to get the answer as 120ft.
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4 - 3x = 8 - 4x please help
Step-by-step explanation:
we switch the like terms across the equation and when we do that we change the signs
the prices of used car tires is a normally distributed variable with a mean of $50 and variance of $100. what is the probability that the price of a used car tire will be less than $65?
The probability that the price of the used car is less than $65 is 0.55962.
Here it is given that the price of the used car has a mean of $50 and a variance of $100.
Since it follows normal distribution X~N(μ, σ²)
where,
μ = 50
σ = 100
Here we need to find the probability P(X < 65)
To find this we need to convert the distribution to the standard normal form, N~(0, 1)
We know,
P(X < (x - μ)/ σ) = Φ(Z)
Therefore we get
P(X < (65 - 50)/ 100)
Solving this we get the Z value
= Φ(0.15)
From the Z table, we can get our p-value.
= 0.55962
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What are the solutions of the quadratic equation?
`2x^{2}+12x+18=0`
show work
The solutions for the quadratic equation:
2x^2 + 12x + 18 = 0
are:
x₋ = -5.55
x₊ = -0.4
How to solve the quadratic equation?
Remember that for a general quadratic equation:
a*x^2 +b*x + c = 0
The solutions are of the form:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex]
Here we have the quadratic:
2x^2 + 12x + 18 = 0
Then using the formula for the solutions we will get:
[tex]x = \frac{-12 \pm \sqrt{12^2 - 4*2*18} }{2*2} \\\\x = \frac{-12 \pm 10.2}{4}[/tex]
The two solutions of the quadratic equation are:
x₋ = (-12 - 10.2)/4 = -5.55
x₊ = (-12 + 10.2)/4 = -0.4
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Which histogram shows a left-skewed distribution? A. A bar chart plots data points (1, 1), (2, 4), (3, 1), (4, 2), (5, 3), (6, 4) and (7, 3). B. A bar chart plots data points (1, 1), (2, 2), (3, 3), (4, 4), (5, 3), (6, 2) and (7, 1 point 5). C. A bar chart plots data points (1, 3), (2, 4), (3, 2), (4, 2), (5, 1), (6, 1) and (7, 1). D. A bar chart plots data points (1, 4), (2, 2), (3, 4), (4, 3), (5, 1), (6, 1) and (7, 2).
The histogram that shows a left-skewed distribution is given by:
A. A bar chart plots data points (1, 1), (2, 4), (3, 1), (4, 2), (5, 3), (6, 4) and (7, 3).
What is an histogram?An histogram is a graph that shows the number of times each element x was observed on the data-set.
Hence, the meaning of a point (x,y) on the histogram is given as follows:
Observation x was observed y times on the data-set.
The skewness of a distribution can be explained as follows:
Left-skewed: The higher values are the ones that are observed the most in the data-set.Symmetric: The middle values are the ones that are observed the most in the data-set.Right-skewed: The lower values are the ones that are observed the most in the data-set.Hence in this problem, for the left-skewed histogram, we want an "increasing" histogram, hence option A is correct.
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help meeeeeeeeeeeeeeeeee please
Answer: 81
Step-by-step explanation:
[tex]v=\sqrt{2gh}\\\\g=32, v=72\\\\\implies 72=\sqrt{2(32)h}\\\\5184=64h\\\\h=81[/tex]
what 5x5(45x5)x3(56x34)
Answer: 32130000
Step-by-step explanation:
Multiply all of your numbers
help meeeeeeeeeeeeeee pleaseee
Answer:
Vertex: (2, -1)
Axis of Symmetry: x=2
Parabola: Downwards
Step-by-step explanation:
coppinger corp. is considering the use of a computer to assign probability distributions to the input variables used in the analysis. the computer would then randomly select input variables from their distributions and calculate npvs based on those selections. what kind of analysis is this? sensitivity analysis simulation analysis scenario analysis
Coppinger Corp is using Scenario Analysis.
Given, that Coppinger Corp is considering the use of a computer to assign probability distributions to the input variables used in the analysis.
The computer would then randomly select input variables from their distributions and calculate npvs based on those selections
Now, A scenario analysis consists of multiple inputs. For eg. a Company may consider how an increase in available fuel can impact their number of goods sold etc.
Here, also Coppinger Corp is assigning multiple values to the computer and getting outputs.
So, this can be the Scenario Analysis.
Hence, Coppinger Corp is using Scenario Analysis.
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which of the following statements is false? group of answer choices older men tend to have lower muscle density, so the correlation between age and muscle density in older men must be negative. older children tend to be taller than younger children, so the correlation between age and height in children must be positive. a researcher finds that the correlation between two variables is close to 0, so the two variables must be unrelated. taller people tend to be heavier than shorter people, so the correlation between height and weight must be positive.
The correct statement would be older men tend to have lower muscle density, so the correlation between age and muscle density in older men must be negative.
Increased time studying sets of tea time is associated with higher G. P.A. (Grade point average) This is positive. More time studying G. P. A. Life expectancy increases as body weight decreases. As the weight goes down the life expectancy goes up.
Muscle Density:Muscle mass (quantity) is highly correlated with survival in hospitalized patients with advanced cancer, whereas muscle density (quality) is associated with patients’ symptoms, health care use, and survival, according to a group of researchers
Correlation:In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
Your body shape changes naturally as you age. You cannot avoid some of these changes, but your lifestyle choices may slow or speed the process.
The human body is made up of fat, lean tissue (muscles and organs), bones, and water. After age 30, people tend to lose lean tissue. Your muscles, liver, kidney, and other organs may lose some of their cells. This process of muscle loss is called atrophy. Bones may lose some of their minerals and become less dense (a condition called osteopenia in the early stages and osteoporosis in the later stages). Tissue loss reduces the amount of water in your
The amount of body fat goes up steadily after age 30. Older people may have almost one third more fat compared to when they were younger. Fat tissue builds up toward the center of the body, including around the internal organs. However, the layer of fat under the skin gets smaller.
The tendency to become shorter occurs among all races and both sexes. Height loss is related to aging changes in the bones, muscles, and joints. People typically lose almost one-half inch (about 1 centimeter) every 10 years after age 40. Height loss is even more rapid after age 70. You may lose a total of 1 to 3 inches (2.5 to 7.5 centimeters) in height as you age. You can help prevent height loss by following a healthy diet, staying physically active, and preventing and treating bone loss.
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What is the value of x in the equation 3/2 (4x – 1) – 3x = 54 –(x + 2)?
A. 1/16
B. 3/16
C. 19/16
D. 3
The solution for x in the equation given as 3/2 (4x – 1) – 3x = 54 –(x + 2) is 13.375
How to determine the solution to the variable k in the expression?From the question, the algebraic expression is given as
3/2 (4x – 1) – 3x = 54 –(x + 2)
There is no constant to multiply or divide in the expression
3/2 (4x – 1) – 3x = 54 –(x + 2)
Open the brackets
So, we have
6x – 3/2 - 3x = 54 - x - 2
Collect the like term in the above equation
This gives
6x - 3x + x = 54 - 2 + 3/2
Evaluate the like term in the above equation
So, we have the following representation
4x = 53.5
Divide through by 4
x = 13.375
The above equation cannot be solved further
Hence, the solution is x = 13.375
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Find a description this equation represents. Of the numbers 1, 2, 3, 4, and 5, which are solutions to the equation?
4x+6=18
A. If you add four to six times a number, you get 18.
B. Four more than six times a number is 18.
C. If you add six to four times a number, you get 18.
D. Six more than four times a number is 18.
E. Six less than four times a number is 18.
F. Four less than six times a number is 18.
G. If you add six times a number to four, you get 18.
H. If you subtract six from four times a number, you get 18.
i just need the answers
Answer:
s is the price of each laptop that gives maximum profit
p is the maximum total profit that can be made.
Step-by-step explanation:
See the attached worksheet for additional explanation. The question asks for the meaning of s (selling price) and p (total profits) at the vertex of the curve. s is in units of price/laptop, while p is in units of total profits. Implicit in this graph is a demand curve (not shown) that illustrates how demand for laptops changes as the price changes. Also not shown, but assumed, is that if the laptops are sold at a low price, the declining profit margins resuce the overall total profits. This graph incorporates the cost and demand imformation in a manner that provides a condensed view of the more important variable: total profits.
Evaluate the expression:
6² x (16 +19) — 10³
36×(35)-1000=1260-1000=260
how to find 3 to the power of x equals 2187?
i need a step by step detail on how to solve it.
Answer: [tex]x=7[/tex]
Step-by-step explanation:
[tex]3^x=2187\\\\x=\log_{3} (2187)\\\\x=7[/tex]
