When p is a prime number greater than 3, the. option that is not a factor of 6p is A. p^2 (B
What is the factor about?The process of factorization simply has to do with the division of arithmetic algebraic expressions into the sum of their factors. in this case, multiplying the variables will yield the original expression. It means the writing of a number into smaller or simpler numbers that give the original number.
6*p / 6*p = 1
6*p / 3*p = 2
6*p / 2*p = 3
6*p / 3 = 2*p
6*p / p^2 = 6 / p which is not divisible by any prime number greater than 3.
The correct option is A.
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Complete question
If p is a prime number greater than 3, which of the following is NOT a factor of 6p ?
(A) p^2 (B) 6p (C) 3p (D) 2p (E) 3
A man bought one almirah for Rs 3640 and the other for Rs 4800. He sold the first almirah for a loss of 15% and the other at a profit of 13 1/3 %. How much did he gain or lose in the whole transaction.
Please tell the answer .
Answer:
Rs. 94
Step-by-step explanation:
We will denote the almirahs as A1 and A2 for ease of explanation. I am omitting currency until the final answer
For the following computations note that x% = x/100 in decimal so
15% = 15/100 = 0.15
13 1/3% = 13 1/3 ÷ 100 = 40/3 ÷ 100 = 40/300
Cost of A1 = 4800
A1 was sold for a loss of 15%
Loss = 15% of 3640 = 0.15 x 3640 = Rs 546
A2 bought at 4800 was sold at a gain of 13 1/3%
Gain from sale = 13 1/3% of 4800 = 40/300 x 4800 =Rs 640
We have to treat gain as positive and loss as negative to arrive at the answer
Net gain/loss = 640 - 546 = Rs. 94
match the given side lengths to the correct inequality that represents the range of the 3rd side. (9 cards will be left over)
A triangle's third side must always be between (but not exactly equal to) the other two sides' total and difference in length. Take the numbers 2, 6, and 7 as an example. In light of this, the third side length must be more than 4 and lower than 8.
What is triangle?Three edges and three vertices make up a triangle, which is a polygon. It is among the fundamental shapes in geometry. Triangle ABC refers to a triangle with the vertices A, B, and C. Any three points in Euclidean geometry that are not collinear determine a singular triangle and a singular plane simultaneously. In geometry, a triangle is a three-sided polygon with three sides, three vertices, and three edges.The fact that a triangle's internal angles can be added up to equal 180 degrees is its most significant characteristic. Triangle's angle sum property is what this characteristic is known as. When three straight lines cross, a triangle is the result. There are always three sides and three corners to a triangle (angles). A triangle's vertex is the location at which two of its sides meet.To learn more about triangle, refer to:
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The dashed triangle is the image of the solid triangle for a dilation centered at the origin. What is the scale factor?
Option(c) is the correct answer i.e. the scale factor of the given triangle is 4/9.
What is Scale factor?
The ratio of the scale of an original thing to a new object that is a representation of it but of a different size is known as a scale factor.
Let's take the measurement of one side of the original triangle and dashed triangle.
The side of Original triangle is 18 units and,
The side of dashed(new) triangle is 8 units
We know that,
Scale factor = dimension of new figure ÷ dimension of original figure
= 8 units / 18 units
= 4/9.
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A dentist was making note of his upcoming appointments with different aged patients and the reasons for their visits.
The probability that an appointment is with a patient under 18 years old is 0.9, the probability that it is for a regular cleaning is 0.15, and the probability that it is with a patient under 18 years old and is for a regular cleaning is 0.14.
What is the probability that a randomly chosen appointment is with a patient under 18 years old or is for a regular cleaning?
Write your answer as a whole number, decimal, or simplified fraction.
The probability that a randomly chosen appointment is with a patient under 18 years old or is for regular cleaning is 0.95.
The probability of a random appointment being with a patient under 18 or for regular cleaning is calculated as P(Under 18 or Regular Cleaning) = P(Under 18) + P(Regular Cleaning) - P(Under 18 and Regular Cleaning).
P(Under 18) = 0.9
P(Regular Cleaning) = 0.15
P(Under 18 and Regular Cleaning) = 0.14
So, P(Under 18 or Regular Cleaning):
= 0.9 + 0.15 - 0.14
Apply the arithmetic operation, and we get
= 0.91.
Thus, the required probability is 0.91.
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Mariah made a cylinder out of clay that had a radius of 6 cm and a volume of 72π cm^3. She painted the entire surface of the cylinder purple. How many square centimeters of the cylinder did Mariah paint in terms of π?
Enter the correct answer in the box/blank in terms of π.
_______ cm^2 (squared)
[tex]Answer:\ \boxed{96\pi }\ cm^2[/tex]
Step-by-step explanation:
[tex]R=6\ cm \ \ \ \ V_c=72\pi \ cm^3\ \ \ \ S_c=?[/tex]
[tex]V_c=\pi R^2H\\[/tex]
[tex]72\pi =\pi 6^2H\\\\72=36H\\[/tex]
Divide both parts of the equation by 36:
[tex]2=H\\\\Thus,\ H=2\ cm[/tex]
[tex]S_c=2(\pi R^2)+2\pi R(H)\\\\S_c=2\pi (6^2)+2\pi 6(2)\\\\S_c=2\pi(36)+12\pi (2) \\\\S_c=72\pi +24\pi \\\\S_c=96\pi\ cm^2[/tex]
Is x = 6 a solution to the equation x + 13 = 18?
A. No, because 6 + 13 = 18 is not true.
B. Yes, because 6 + 13 = 18 is not true.
C. No, because 6 + 13 = 18 is true.
D. Yes, because 6 + 13 = 18 is true.
On solving the provided question, we can say that x = 6 a solution to the equation x + 13 = 18, no, because 6 + 13 = 18 is not true, as 6 + 13 = 19
What is equation?A mathematical equation is a formula that joins two statements and uses the equal symbol (=) to indicate equality. A mathematical statement that establishes the equality of two mathematical expressions is known as an equation in algebra. For instance, in the equation 3x + 5 = 14, the equal sign places the variables 3x + 5 and 14 apart. The relationship between the two sentences on either side of a letter is described by a mathematical formula. Often, there is only one variable, which also serves as the symbol. for instance, 2x – 4 = 2.
x = 6 a solution to the equation x + 13 = 18
No, because 6 + 13 = 18 is not true, as 6 + 13 = 19
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the first three steps for completing the square to solve a quadratic equation are shown. x squared minus 8 x plus 2 equals 0. x squared minus 8 x equals negative 2. x squared minus 8 x plus box equals negative 2 plus box.questionwhat number goes in the boxes to complete the third step?
The number that goes in the box is 4, since 4 + (-2) = 2, which is the coefficient of x^2 in the equation. The third step is to add the same number to both sides of the equation, so the box needs to contain 4.
To complete the third step in solving a quadratic equation using the completing the square method, the number that goes in the box is 4. This is because the goal is to make the coefficient of x^2 on the left side of the equation equal to the coefficient of x^2 on the right side. The equation is x^2 - 8x + 2 = 0, so the coefficient of x^2 on the left side is 1, and the coefficient of x^2 on the right side is 2. Therefore, the number that needs to be added to the left side is 1, and the number that needs to be added to the right side is also 1. This means that 4 needs to be added to both sides of the equation, so 4 is the number that goes in the box.
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Please help!! I can’t figure this out
Answer:
$3213
Step-by-step explanation:
You want to know the total spent on rent and food if rent is 7/20 of your budget and food is 1/2 of your budget of $3780 per month.
AmountsThe amount spent on each item is the product of the fraction of the budget it is and the total amount being budgeted (your income).
rent = 7/20 × 3780 = 1323
food = 1/2 × 3780 = 1890
The total spent on these items is the sum of these amounts:
rent + food = 1323 +1890 = 3213
A total of $3213 is budgeted for rent and food.
__
Additional comment
Your calculator can figure this handily.
You can factor out the income amount and add the fractions first:
(7/20·3780) +(1/2·3780) = 3780(7/20 +1/2) = 3780(7/20 +10/20)
= 3780(17/20) = 3213
Examine the number line,and then choose the correct symbol to complete each statement
After examine the number line,and then the correct symbol to complete each statement is -2/4<-1/4; 3/4>2/4;-1/4<0.
What is Number line ?
A number line is a diagram of a graduated straight line used to represent real numbers in introductory mathematics. It is assumed that every point on a number line corresponds to a real number, and that every real number corresponds to a point.
What is Symbol?
A mark, sign, or term that denotes, denotes, or is taken to denote an idea, an item, or a relationship is known as a symbol. By connecting seemingly unrelated ideas and events, symbols help people look beyond the known and the visible.
In the question given some relation between the real number which are
a) -2/4 < -1/4 because negative number has the property that the those numbers which is smaller is always greater.
b) 3/4 > 2/4 both numbers are positive so clearly it 3/4 comes after 2/4 so it is greater.
c) -1/4 < 0 zero is always greater than the negative numbers
So, after examine the number line,and then the correct symbol to complete each statement is -2/4<-1/4; 3/4>2/4;-1/4<0.
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2. If -2 is the root of a quadratic equation x²-5x+c=0, find
a: the value of c
b: the other root
The value of c according to the given equation as described is; -14.
The other root of the equation in discuss is; 7.
What is the value of c and determine the other root?As evident in the task content; if -2 is the root of a quadratic equation x²-5x+c=0, then x = -2 holds true for the equation; therefore, we have;
a). (-2)² - 5 (-2) + c = 0
4 + 10 + c = 0
c = -14
Hence, the value of c is; -14.
b). By substituting into the equation; we have;
x² - 5x - 14 = 0
(x - 7) (x + 2) = 0
x = 7 OR x = -2.
Therefore, the other root of the equation is; 7.
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Which operation results in a binomial
A. -
B. +
C. •
Answer:
answer is d
Step-by-step explanation:
because of binomials connect to the plus sign but which solves the equations of the multiplicationsFind the equation of the ellipse with the following properties.
The ellipse with foci at (4, 0) and (-4, 0); y-intercepts (0, 3) and (0, -3)
Step-by-step explanation:
The equation of the ellipse
[tex]\displaystyle\\\boxed {\frac{x^2}{a^2} +\frac{y^2}{b^2} =1}[/tex]
[tex]F^2=a^2-b^2[/tex]
Given: F₁(-4,0) F₂(4,0) b₁(0,-3) b₂=(0,3)
[tex]\displaystyle\\a^2=F^2+b^2\\\\a^2=4^2+3^2\\\\a^2=16+9\\\\a^2=25\\\\a_{1,2}=б5\\\\So,\ a_1(-5,0)\ \ \ \ a_2(5,0)\\\\Thus,\ \frac{x^}{(б5)^2} +\frac{y^2}{(б3)^2} =1\\\\\frac{x^2}{25}+\frac{y^2}{9} =1\ \ \ \ -\ the\ equation \ of\ the\ ellipse[/tex]
three friends went to an icecream store and tried a new flavor the store was testing. After
trying the flavor, they were asked to rate the icecream on 9-point rating scale ranging from
1-Hate it to 9-Love it. The friends ratings were: Destiny 9, Taylor 6, Jeremiah 5. What is the
mean of their ratings?
The mean of their ratings is 7.
How to calculate the mean of their ratings?
Mean (also called the average) is used to describe the central tendency of a set of numerical data. It is calculated by summing up all the values in the data set and dividing by the number of values.
Since We have the ratings:
Destiny 9, Taylor 6, Jeremiah 5
Mean = (9 + 6 + 5)/3 = 20/3 = 7
Note: we divide by 3 because there are 3 people
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Benjamin gets to go to festival with his friend for $120 The cost covers $45 for his lodging and 2 tickets to the festival, including all of his meals.
How much is a ticket to the festival?
Answer:
is $27.50 per each teacket
Please Help ASAP - 100pts + Brainliest!
Answer:
g(x) = e^(x+2) -3
Step-by-step explanation:
You want the transformed function g(x) of the parent function f(x) = e^x given that g(x) has a horizontal asymptote of -3 and goes through the point (-2, -2).
TranslationThe parent function goes through the point (0, 1), which has apparently been translated to (-2, -2) by the transformation. That translation looks like ...
g(x) = f(x -h) +k . . . . . . . . f(x) translated h units right and k units up
ApplicationOur function has been translated from 0 to -2 in the horizontal direction, or left 2 units: h = -2.
It has been translated from 1 to -2 in the vertical direction, or down 3 units: k = -3.
Then the translated function is ...
g(x) = f(x +2) -3
g(x) = e^(x+2) -3
can you include the full picture?
Consider the following data: 13,11,5,2,9,14 Step of 3 : Calculate the value of the sampe variance Round your answer to one decimal place.Refer to Exercise 3. Calculate and interpret the standard deviation of the random variable X.
The sample variance of the given data 13,11,15,2,9,14 is 356.6.
Sample variance can be defined as the expectation of the squared difference of data points from the mean of the data set. It is an absolute measure of dispersion and is used to check the deviation of data points with respect to the data's average.
we have data 13, 11, 5, 2, 9, 14, and we have to determine the sample variance of given data.
Sample variance is given by,
[tex]S^2=\frac{\sum^n_1(X_i-X')^2}{(n-1)}[/tex]
where X' is the sample mean,
the mean of given data is,
[tex]X'=\frac{(13+11+5+2+9+14)}{6} \\\\X'=\frac{54}{6} \\\\X'=9[/tex]
now the sample variance is given by,
[tex]S^2=\frac{(13+9)^2+(11+9)^2+(5+9)^2+(9+9)^2+(14+9)^2}{6-1} \\\\S^2=\frac{(484+400+289+81+529)}{5} \\\\S^2=356.6[/tex]
Hence the sample variance is 356.6.
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what is the slope of the line through the points (-2,9) and (-4,6) ?
Answer:
The slope is [tex]\frac{3}{2}[/tex]
Step-by-step explanation:
The slope formula would be: [tex]\frac{y2-y1}{x2-x1}[/tex]
So for (-2,9) and (-4,6), we can plug in the variables: (x1, y1) and (x2, y2)
Let's use the numbers in the equation provided above:
[tex]\frac{6-9}{-4-(-2)}[/tex]
= [tex]\frac{-3}{-2}[/tex]
= [tex]\frac{3}{2}[/tex]
Answer:
3/2
Step-by-step explanation:
[tex]\frac{y2-y1}{x2-x1} = \frac{6-9}{-4-(-2)} = \frac{-3}{-2} =\frac{3}{2}[/tex]
State the domain and range for the following relation. Then determine whether the relation represents a function. {( -4,6), (-4,7), (6,5), (7,9)} The domain of the relation is __. (Use a comma to separate answers as needed.) The range of the relation is ___ (Use a comma to separate answers as needed.) Does the relation represent a function? Choose the correct answer below. A. The relation is a function because there are no ordered pairs with the same second element and different first elements. B. The relation is a function because there are no ordered pairs with the same first element and different second elements. C. The relation is not a function because there are ordered pairs with 5 as the second element and different first elements. D. The relation is not a function because there are ordered pairs with - 4 as the first element and different second elements.
Since the input value of domain -4 has two different output values (7 and 6), the relation is not a function.
By definition, a relation is a function if each input value has only one output value.
Given the relation:
(-4,6)
(-4,7)
(6,5)
(7,9)
The domain is the set of the x-coordinates of each ordered pair (You do not need to write 4 twice):
domain ={-4 ,6,5}
The range is the set of the y-coordinates of each ordered pair :
range= {6,7,5,9}
Since the input value 4 has two different output values (7 and 6), the relation is not a function.
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You pick a card at random. Without putting the first card back, you pick a second card at random. 1 2 3 4 What is the probability of picking a 2 and then picking a 1? Write your answer as a fraction or whole number.
The probability of picking a 2 and then picking a 1 would be = 1/2
What is probability?Probability is definitely as the expression that to represents the number of outcomes of an event.
The number of cards such as 1,2,3,4 = 4 cards.
The cards picked at random = a 1 and a 2
Therefore, the probability = 2/4 = 1/2
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Tomhas$84,076inasavingsaccountthatearns15%annually.Theinterestisnotcompounded.Tothenearestdollar,howmuchinterestwillheearnin6months?
Use the formula i = prt, where i is the interest earned, p is the principal (starting amount), r is the interest rate expressed as a decimal, and t is the time in years.
$75668.4 is the interest earned in 6 months.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Tom has $84,076 in a savings account that earns 15% annually.
The interest is not compounded.
We have to find the interest he earned in 6 months.
Use the formula i = prt,
where i is the interest earned,
p is the principal (starting amount),
r is the interest rate expressed as a decimal,
and t is the time in years.
i=84076×15/100×6
=84076×0.15×6
=$75668
Hence, $75668.4 is the interest earned in 6 months.
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Determine whether the sequence converges or diverges. If it converges, find the limit. an=(1+2/n)^n
This sequence converges to e. The sequence (1 + 2/n)^n is a geometric sequence in which each successive term is found by multiplying the previous term by a constant.
This sequence is an example of a geometric sequence where each successive term is found by multiplying the previous term by a constant. In this case, the constant is equal to (1 + 2/n).
By using the limit definition of a geometric sequence, we can see that the limit of this sequence is given by lim n→∞ (1 + 2/n)^n = e. Therefore, the sequence converges to e.
The sequence (1 + 2/n)^n converges to e.
The sequence (1 + 2/n)^n is a geometric sequence in which each successive term is found by multiplying the previous term by a constant. By using the limit definition of a geometric sequence, it can be determined that the limit of this sequence is given by lim n→∞ (1 + 2/n)^n = e. As a result, the sequence converges to e.
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For the function given below, find a formula for the Riemann sum obtained by dividing the interval [a,b] into n equal subintervals and using the right-hand endpoint for each [tex]c_k[/tex]. Then take a limit of this sum as [tex]n[/tex]→∞ to calculate the area under the curve over [a,b].
[tex]f(x)=2x[/tex] over the interval [1,5]
The Riemann sum formula is ∫(1,5) 2x = 24
How to determine the Riemann sum formulaFrom the question, we have the following parameters that can be used in our computation:
f(x) = 2x over the interval [1, 5]
The Riemann sum for f(x) = 2x over the interval [1,5] using the right-hand endpoint for each subinterval is given by the formula:
(Δx/2) * (2c1 + 2c2 + ... + 2cn) where c1, c2, ..., cn are the right-hand endpoints of the subintervals and Δx = (b - a)/n.
As a general rule:
As n approaches infinity, the width of the subintervals approaches zero,
Also, we have the Riemann sum approaches the definite integral of f(x) = 2x over the interval [1,5], which is x² evaluated at the interval's endpoints.
So, the formula becomes
∫(1,5) 2x = x²
Substitute the intervals
∫(1,5) 2x = 5² - 1²
Evaluate
∫(1,5) 2x = 24
Hence, the formula is ∫(1,5) 2x = 24
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LA REPRESA POZA HONDA ALMACENA 11 MILLONES DE METROS CÚBICOS DE AGUA. ¿CUÁNTOS KILÓMETROS DE AGUA ALMACENA?
Porfavor ayudenme es para Matematicas
Answer: 11/1000 o 1.1%
Step-by-step explanation:
11 ÷ 1000
0.011
11/1000 o 1.1%
The article "Kids Digital Day: Almost 8 Hours" (USA Today, January 20, 2010) summarized results from a national survey of 2002 Americans age 8 to 18. The sample was selected in a way that was expected to result in a sample representative of Americans in this age group.
a. Of those surveyed, 1321 reported owning a cell phone. Use this information to construct and interpret a 90% confidence interval estimate of the proportion of all Americans age 8 to 18 who own a cell phone.
b. Of those surveyed, 1522 reported owning an MP3 music player. Use this information to construct and interpret a 90% confidence interval estimate of the proportion of all Americans age 8 to 18 who own an MP3 music player.
c. Explain why the confidence interval from Part (b) is narrower than the confidence interval from Part (a) even though the confidence level and the sample size used to compute the two intervals was the same.
The confidence interval from Part (b) is narrower than the confidence interval from Part (a) because the proportion of Americans age 8 to 18 who own an MP3 music player is higher than the proportion of Americans age 8 to 18 who own a cell phone.
a. 90% confidence interval estimate of the proportion of all Americans age 8 to 18 who own a cell phone:
(1321/2002) ± (1.645*sqrt( (1321/2002)*(1-(1321/2002)) / 2002)) = (0.6608 ± 0.0402)
Interpretation: We are 90% confident that the true proportion of all Americans age 8 to 18 who own a cell phone is between 0.6206 and 0.7010.
b. 90% confidence interval estimate of the proportion of all Americans age 8 to 18 who own an MP3 music player:
(1522/2002) ± (1.645*sqrt( (1522/2002)*(1-(1522/2002)) / 2002)) = (0.7610 ± 0.0346)
Interpretation: We are 90% confident that the true proportion of all Americans age 8 to 18 who own an MP3 music player is between 0.7264 and 0.7956.
c. The confidence interval from Part (b) is narrower than the confidence interval from Part (a) because the proportion of Americans age 8 to 18 who own an MP3 music player is higher than the proportion of Americans age 8 to 18 who own a cell phone. This means that the sample size is more likely to result in a more accurate estimation of the true population proportion for the MP3 music player than for the cell phone.
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A kite is flying 20 ft off the ground. It’s line is pulled taut and casts a 15 feet shadow
Answer:ts line is pulled taut and casts a 15 ft - shadow. Find the length of the line. If necessary, round your answer to the nearest tenth.
Step-by-step explanation:
The polygon in the diagram is a square with center P. The length from the center to the vertex is [tex]6\sqrt{2} in[/tex]. Find the area of the square to the nearest tenth of an inch.
A) 24.0 [tex]in^{2}[/tex]
B) 72.0 [tex]in^{2}[/tex]
C) 36.0 [tex]in^{2}[/tex]
D) 144.0 [tex]in^{2}[/tex]
The area the polygon in the diagram with, a length from the center to the vertex 6√2 in is 144.0 in².
What is polygon?Any closed curve that consists entirely of connected line segments (sides) without any intersections is known as a polygon in geometry. The three simplest polygons are pentagons, triangles with three sides, and quadrilaterals with four sides (five sides). The two types are concave and convex polygons.
Convex polygons are those without any extended sides crossing over them. Equilateral refers to a polygon with equal sides. Interior angles of equiangular shapes are equal. Every equilateral and equiangular polygon is a regular polygon (e.g., an equilateral triangle or a square).
As all sides are equal, all diagonals are equal too, and the angle of adjacent diagonals are 90° with vertical opposite angle.
Therefore,
Each side = [tex]\sqrt{\text {diagonal}^2+ \text {diagonal}^2}[/tex]
Each side = [tex]\sqrt{(6\sqrt{2} )^2+(6\sqrt{2} )^2}[/tex]
Each side = [tex]\sqrt{(36\times 2+36\times 2}[/tex]
Each side = [tex]\sqrt{144}[/tex]
Each side = 12 inch
Area = 12 × 12
= 144.0 in²
Thus, the area the polygon in the diagram with, a length from the center to the vertex 6√2 in is 144.0 in².
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You buy 4 small candles and 2 large candles for $14. Your friend buys 5 small candles and 3 large candles for $20. What is the cost of each small candle? of each large candle?
Each small candle costs $_ and each large candle costs $_.
Using simultaneous equations, each small candle costs $1.00 and each large candle costs $5.00.
What are simultaneous equations?Simultaneous equations are two or more equations solved concurrently.
Simultaneous equations are also known as a system of equations.
Small Candle Large Candle Total Cost
You buy 4 2 $14
A friend buys 5 3 $20
Let the cost of the small candle each = s and the cost of the large candle each = l
Equations:Equation 1: 4s + 2l = 14
Equation 2: 5s + 3l = 20
Multiply Equation 1 by 1.5:
Equation 3: 6s + 3l = 21
Subtract Equation 2 from Equation 3:
6s + 3l = 21
-
5s + 3l = 20
s = 1
= $1
From Equation 1:
4s + 2l = 14
4 + 2l = 14
2l = 10
l = 5
= $5
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Emily was offered a job after college earning a salary of $35,000. She will get a raise
of $2,000 after each year working for the company. Answer the questions below
regarding the relationship between salary and the number of years working at the
company.
, and the
The independent variable, x, represents the
dependent variable is the
depends on the
because the
A function relating these variables is C(x)
So C(4)
meaning 4
Answer:
The independent variable, x, represents the number of years working at the company. The dependent variable, C(x), represents the salary. The salary depends on the number of years working at the company because the salary increases by $2,000 for each year of work. A function relating these variables is C(x) = 35,000 + 2,000x, where x is the number of years working at the company and C(x) is the salary. So C(4) = 35,000 + 2,000(4) = $43,000 meaning that if Emily works for 4 years, her salary will be $43,000.
Need help with this equation
4x⁴ +12x³ +15x² +9x −6 is the expression which can represent function f(g(x)). Thus, option D is correct.
What is function?A rule that transforms one number into another is called a function. However, not every rule outlines an appropriate function. In addition to introducing some of the mathematical terms related to functions, this unit explains how to determine if a given rule describes a valid function.
Given
f(x) = x² + 3x - 6
g(x) = 2x² +3x
To find expression for f(g(x))
Put the value of g(x) place of x, f(x)
f(x) = x² + 3x - 6
(2x² +3x)² + 3(2x² +3x) - 6
4x⁴ +12x³ +15x² +9x −6
Thus, 4x⁴ +12x³ +15x² +9x −6 is the expression which can represent f(g(x)).
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Highway Design A section of highway connecting two hillsides with grades of 6% and 4% is to be built between two points that are separated by a horizontal distance of 2000 feet (see figure). At the point where the two hillsides come together, there is a 50-foot difference in elevation.
(a) Design a section of highway connecting the hillsides modeled by the function {eq}f(x) = ax^3 + bx^2 + cx + d (-1000 \le x \le 1000) {/eq}. At the points A and B, the slope of the model must match the grade of the hillside.
(b) Use a graphing utility to graph the model.
(c) Use a graphing utility to graph the derivative of the model.
(d) Determine the grade at the steepest part of the transitional section of the highway.
You need to find the four constants, a, b, c, and d. At x = -1000, you have a value for f(x).
A)Outline it:
f(-1000) = -1000 = a (-1000)^3 b(-1000)^2 + c (-1000) + d
At x = 1000, you have a value for f(x).
B)Outline it:
f(+1000) = a (1000)^3 + b (1000)^2 + c (1000) + d
You know the slope's value, f'(x), and when x = -1000, f'(-1000) equals 3 a (-1000)2 + 2 b (-1000) + c.
At x = + 1000 f'(1000), you have a value for the slope: 3 a (1000)2 + 2 b (1000) + c
C) That is four linear equations with a, b, c, and d as the four unknowns.
When you simultaneously solve all four equations, you will get the function and its derivative, which you may graph.
D) The steepest part is where the derivative's absolute value is at its highest.
You can find that extreme by setting the derivative of the derivative equal to zero.
6 a x + 2 b = 0
solve for x and calculate the slope at that point.
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