Exponential Decay
The model for the exponential decay of a quantity Mo is:
[tex]M=M_o\cdot e^{-\lambda t}[/tex]Where λ is a constant and t is the time.
The half-life of a radioactive isotope is the time it takes to halve its initial mass. It can be calculated by making M = Mo/2 and solving for t:
[tex]\begin{gathered} \frac{M_o}{2}=M_o\cdot e^{-\lambda t} \\ \text{Simplifying:} \\ e^{-\lambda t}=\frac{1}{2} \\ \text{Taking natural log:} \\ -\lambda t=-\log 2 \\ t=\frac{\log 2}{\lambda} \end{gathered}[/tex]It's required to calculate the remaining mass of an isotope of Mo = 175 gr after 5 half-lives have passed, that is. we must calculate M when t is five times the value calculated above.
Substituting in the model:
[tex]M=175gr\cdot e^{-\lambda\cdot\frac{5\log 2}{\lambda}}[/tex]Simplifying (the value of λ cancels out):
[tex]\begin{gathered} M=175gr\cdot e^{-5\log 2} \\ \text{Calculating:} \\ M=175gr\cdot0.03125 \\ M=5.46875gr \end{gathered}[/tex]Rounding to the nearest gram, 5 grams of the radioactive isotope will be left after the required time.
Real estate prices in a Denver neighborhood areNormally distributed with a mean price of $187,500 and astandard deviation of $12,500.ALDenver Neighborhood Real Estate Pricing
First, find the z-score of the two values $150,000 and $225,000
[tex]\begin{gathered} \text{z-score for }x_1 \\ z=\frac{x-\mu}{\sigma} \\ z=\frac{150000-187500}{12500} \\ z=\frac{-37500}{12500} \\ z_1=-3 \\ \; \\ \text{z-score for }x_2 \\ z=\frac{x-\mu}{\sigma} \\ z=\frac{225000-187500}{12500} \\ z=\frac{37500}{12500} \\ z_2=3 \end{gathered}[/tex]Since the z-scores are both 3 standard deviations away from the mean, by Emperical rule, we conclude that about 99.7% of the homes will be priced between $150,000 and $225,000.
need help finding the exact value of sec pi/3
Solution:
Given:
[tex]sec(\frac{\pi}{3})[/tex]To find the exact value,
Step 1: Apply the trigonometri identieties.
From the trigonometric identities,
[tex]sec\text{ }\theta\text{ =}\frac{1}{cos\theta}[/tex]This implies that
[tex]sec(\frac{\pi}{3})=\frac{1}{\cos(\frac{\pi}{3})}[/tex]Step 2: Evaluate the exact value.
[tex]\begin{gathered} since \\ \cos(\frac{\pi}{3})=\frac{1}{2}, \\ we\text{ have} \\ sec(\frac{\pi}{3})=\frac{1}{\cos(\pi\/3)}=\frac{1}{\frac{1}{2}}=2 \end{gathered}[/tex]Hence, te exact value of
[tex]sec(\frac{\pi}{3})[/tex]is evaluated to be 2
Help me due is tomorrow
Step-by-step explanation:
5.3g+9=2.3g+15
5.3g-2.3g=15-9
3g=6
3g/3=6/3
g=2
B,5.3(2)+9=2.3(2)+15
10.6+9=4.6+15
19.6=19.6
g = 2
Step-by-step explanation:5.3g + 9 = 2.3g + 15
Subtract 9 from both sides.
5.3g + 9 - 9 = 2.3g + 15 - 9
5.3g = 2.3g + 6
Subtract 2.3g from both sides
5.3g - 2.3g = 2.3g - 2.3g + 6
3g = 6
Divide both sides by 3
g = 2
To check if the value of g is correct, substitute the value of g in the equation above and remember that the both sides should be equal because of the equal sign (=) in the equation.
5.3g + 9 = 2.3g + 15
5.3(2) + 9 = 2.3(2) + 15
10.6 + 9 = 4.6 + 15
19.6 = 19.6
Which of the following is not a valid way of starting the process of factoring60x² +84x +49?Choose the inappropriate beginning below.O A. (x )(60)OB. (2x (30%)O C. (6x X10x)OD. (2x (5x )
Given the equation:
60x^2 + 84x + 49
We are to determine among the options which is not a process of factorizing.
In factorizing, you get factors of the given numbers of the equation that when they are being multiplied or added, they give the numbers in the equation.
So, looking at the options, the only option that does not satisfies the requirement for starting a factorization process is B, which is (2x (30%)
Therefore, the inappropriate process of starting factorization among the option is option B which is (2x (30%).
find the inverse function of g(x)= x-1÷x+5
1. replace g(x) with y:
[tex]y=\frac{x-1}{x+5}[/tex]2.Replace every x with a y and replace every y with an x
[tex]x=\frac{y-1}{y+5}[/tex]3. Solve for y:
[tex]\begin{gathered} (y+5)x=y-1 \\ yx+5x=y-1 \\ yx-y=-1-5x \\ y(x-1)=-1-5x \\ y=\frac{-1-5x}{x-1} \end{gathered}[/tex]4. Replace y with g−1(x) g− 1 ( x ):
[tex]g(x)^{-1}=\frac{-5x-1}{x-1}[/tex]What is the equation of the line that passes through points (1,-19) and (-2,-7)?
This problem is about linear equations. We need to find the equation of the line w
Wayne is hanging a string of lights 89 feet long around the three sides of his rectangular patio, which is adjacent to his house. The length of his patio, the side along the house, is 5 feet longer than twice its width. Find the length and width of the patio.
SOLUTION:
Case: Rectangles
Method:
The sum is:
w + w + 2w + 5 = 89
4w + 5 = 89
4w = 89 - 5
4w = 84
w = 84/ 4
w= 21 feet
The length, l
l = 2w + 5
l = 2( 21) + 5
l = 42 + 5
l = 47 feet
Final answer:
length of the patio, l = 47 feet
width of the patio, w= 21 feet
The diamond method for factoring: Fill in the missing value
Consider a quadratic expression, let "m" and "n" represent the factors.
The diamond method of factoring is the following:
On the left of the diamond, there is one of the factors, for example, "m", of the right of the diamond you will find the other factor "n".
On the top of the diamond, you will find the product of both factors, on the bottom of the diamond you will find the sum of the factors.
Looking at the given diamond, you know the result of the product and the sum of both factors:
[tex]m*n=-15[/tex][tex]m+n=14[/tex]Using these expressions, you can find both factors.
- First, write the second expression for one of the variables, for example, for "n"
[tex]\begin{gathered} m+n=14 \\ m=14-n \end{gathered}[/tex]- Second, replace the expression obtained on the second equation:
[tex]\begin{gathered} m*n=-15 \\ (14-n)n=-15 \end{gathered}[/tex]Distribute the multiplication
[tex]14n-n^2=-15[/tex]Zero the expression and order the terms from greatest to least:
[tex]\begin{gathered} 14n-n^2+15=-15+15 \\ 14n-n^2+15=0 \\ -n^2+14n+15=0 \end{gathered}[/tex]- Third, use the quadratic expression to determine the possible values of n:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]Where
a is the coefficient of the quadratic term
b is the coefficient of the x-term
c is the constant
For the quadratic expression obtained, where "n" represents the x-variable.
[tex]-n^2+14n+15=0[/tex]The coefficients are:
a= -1
b=14
c=15
[tex]\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ n=\frac{-14\pm\sqrt{14^2-4*(-1)*15}}{2*(-1)} \\ n=\frac{-14\pm\sqrt{196+60}}{-2} \\ n=\frac{-14\pm\sqrt{256}}{-2} \\ n=\frac{-14\pm16}{-2} \end{gathered}[/tex]Solve the sum and difference separately to determine both possible values for "n"
→Sum:
[tex]\begin{gathered} n=\frac{-14+16}{-2} \\ n=\frac{2}{-2} \\ n=-1 \end{gathered}[/tex]→Difference:
[tex]\begin{gathered} n=\frac{-14-16}{-2} \\ n=\frac{-30}{-2} \\ n=15 \end{gathered}[/tex]- Finally, determine the possible value/s of m:
For n=-1
[tex]\begin{gathered} m+n=14 \\ m+(-1)=14 \\ m-1=14 \\ m=14+1 \\ m=15 \end{gathered}[/tex]For n=15
[tex]\begin{gathered} m+n=14 \\ m+15=14 \\ m=14-15 \\ m=-1 \end{gathered}[/tex]So, the factors are -1 and 15 and the diamond is:
1(c). What is a better deal? Explain. Deal 1: 2 mediums 14'' (round) pizza for $14 total Deal 2: 1 large 20'' (round) pizza for $13 total
To get the better deal of the two, we need to find the cost per area of pizza for each deal and compare.
Deal 1: 2 medium 14'' (round) pizza for $14 total
The area of a circle is calculated as
[tex]A=\pi r^2[/tex]where r is the radius.
The area of the pizza is calculated to be:
[tex]\begin{gathered} r=14 \\ \therefore \\ A_1=\pi\times14^2=196\pi \end{gathered}[/tex]Hence, the total area for the two pizzas will be:
[tex]\Rightarrow196\pi\times2=392\pi[/tex]The cost per square inch of pizza is, therefore, calculated to be:
[tex]\Rightarrow\frac{14}{392\pi}=0.011[/tex]The pizza costs $0.011 per square inch.
Deal 2: 1 large 20'' (round) pizza for $13 total
The area of the pizza is calculated to be:
[tex]\begin{gathered} r=20 \\ \therefore \\ A_2=\pi\times20^2=400\pi \end{gathered}[/tex]Hence, the cost per square inch of pizza is calculated to be:
[tex]\Rightarrow\frac{13}{400\pi}=0.010[/tex]The pizza costs $0.010 per square inch.
CONCLUSION:
The better deal will be the deal with the lesser cost per square inch. As can be seen from the calculation, both deals are about the same price per square inch if approximated. However, without approximation, Deal 2 has a slightly lesser cost per square inch.
Therefore, DEAL 2 IS THE BETTER DEAL.
Let set E be defined as follows:
E = {english, math, french, art}
Which of the following are subsets of set
E
The subsets of E is all the above .
What are subsets of set ?If every component present in Set A is also present in Set B, then Set A is said to be a subset of Set B. To put it another way, Set B contains Set A. As an illustration, if set A has the elements X, Y, and set B contains the elements X, Y, and Z, then set A is the subset of set B.
If every element in a set A is also an element in a set B, then the set A is a subset of the set B. The set A is therefore contained within the set B. AB is used to represent the subset connection. For instance, if the sets A and B are equal, AB but BB, respectively.
Let the event E = {english, math, french, art}
The subsets of E is all the above .
null set is also subset of E
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7x - 15 < 48. Elrich planted seeds from each of x different seed packets in his garden. To plant these seeds, he had to remove plants that were already in the garden. Taking into account the plants he removed and the seeds he planted, he expected to have (select) plants in the garden. From how many different seed packets did Elrich recently plant seeds?
Elrich planted 7 seeds from each of x different seed packets in his garden. To plant these seeds, he had to remove 15 plants. that were already in the garden. Taking into account the plants he removed and the seeds he planted, he expected to have less than equal to 48 plants in the garden.
The inequality :
[tex]7x-15\leq48[/tex]Simplify for x:
[tex]7x-15\leq48[/tex]
Yes or no to tell wether the fact the fraction is equivalent to this decimal __(4.05)_____________________________________Is the following fractions equal to the one decimal listed? 405/99401/9981/33802/198
802/198 is equal to 4.0505
у A 5 8 106 С C m2l= m22= m23= mZ4= m25= needing quadrilaterals area
Angles in a quadrilaterals
The sum of all interior angles in a quadrilateral is 360°
Angle 5 is congruent with angle of 106°
Thus measure of 5 = 106°
These two angles add up to 212°. The remaining to reach 360° is:
360° - 212° = 148°
Angles 1, 2, 3, and 4 are congruent, thus the measure of each one of them is 148/4=37°. Thus
measure of 1 = measure of 2 = measure of 3 = measure of 4 = 37°
5)Which of the following is a critical number of the inequality x^2+5x-6<0 ?
Answer:
B. 1
Explanation:
Given the inequality:
[tex]x^2+5x-6<0[/tex]To find the critical number, first, change the inequality sign to the equality sign :
[tex]x^2+5x-6=0[/tex]Next, solve for x:
[tex]\begin{gathered} x^2+6x-x-6=0 \\ x(x+6)-1(x+6)=0 \\ (x-1)(x+6)=0 \\ x-1=0\text{ or }x+6=0 \\ x=1\text{ or }x=-6 \end{gathered}[/tex]Therefore, from the options, 1 is the critical number.
The correct option is B.
Could you help me with this problem?There are 7 acts in a talent show.A comedian, a guitarist, a magician, a pianist, a singer, a violinist, and a whistler.A talent show host randomly schedules the 7 acts.Compute the probability of each of the following events.Event A: The magician is first, the comedian is second, and the whistler is third.Event B: The first three acts are the guitarist, the pianist, and the singer, in any order.Write your answers as fractions in simplest form.
EXPLANATION
For the event B, the order of the first 3 acts doesn't matter.
So, the number of acts taken from the seven acts when the order doesn't matter is calculated using combinations.
[tex]C(m,n)=\frac{m!}{n!(m-n)!}[/tex][tex]C(7,3)=\frac{7!}{3!(7-3)!}=\frac{7!}{3!4!}[/tex]Computing the factorials:
[tex]C(7,3)=\frac{5040}{6\cdot24}=\frac{5040}{144}=35[/tex]Hence, the number of ways the three acts could be given are 1:C(7,3)
Therefore, the probability of the event B is:
[tex]P(B)=\frac{1}{35}[/tex]For the event A, the order matters, so the difference between combinations and permutations is ordering. When the order matters we need to use permutations.
The number of ways in which four acts can be scheculed when the order matters is:
[tex]P(m,n)=\frac{m!}{(m-n)!}[/tex][tex]P(m,n)=\frac{7!}{(7-3)!}=\frac{5040}{24}=210[/tex]The number of ways the magician is first, the comedian is second and the whistler is third are 1:P(7,4)
Therefore, the probability of the event A is.
[tex]P(A)=\frac{1}{210}[/tex]A tutoring service charges an initial consultation fee of $50 plus $25 for each tutoringsession.A. Write an equation that determines the total cost of tutoring services (y) based on thenumber of tutoring sessions (x).B. If a student decides to purchase 8 tutoring sessions, what will be his total cost?c. If a student had a total cost of $200, how many tutoring sessions did he attend?EditVioInsertFormatThols Table
A. y = 50 + 25x
B. number of session (x) = 8
Substitute x= 8 in the equation y= 50 + 25x
y = 50 + 25( 8 )= 50 + 200 = $250
The total cost for 8 tutoring sessions is $250
C. y = $200
x= ?
y = 50 + 25x
200 = 50 + 25x
200 - 50 = 25x
150 = 25x
Dividing through by 25
x = 150/25 =6
He attended 6 tutoring sessions
determine whether the given by binomial is a factor of the polynomial p(x) . If so, find the remaining factors of p(x).
The given binomial is a factor of the polynomial p(x) with remaining factors of (x + 1)(x - 1).
What is termed as the factors of polynomial?Factorisation is the process of determining the factors of a given value as well as mathematical expression. Factors are integers which are multiplied together to create the original number.For the given question.
The polynomial is given as; x³ + 2x² -x - 2.
The binomial is given as; (x +2).
The, to get the remainder, divide the polynomial with the binomial.
= (x³ + 2x² - x - 2)/ (x +2)
Taking x² common from the first two terms of the numerator and (-1) from the last two terms.
= x²(x + 2) - (x + 2)/ (x +2)
Taking (x + 2) common from two terms.
= (x + 2)(x² - 1)/(x + 2)
Cancel (x + 2) from both.
= (x² - 1)
Now use the identity to open the square.
(a² + b² ) = (a + b) (a - b)
= (x + 1)(x - 1).
Thus, the given binomial is a factor of the polynomial p(x) with remaining factors of (x + 1)(x - 1).
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The correct question is-
Determine whether the given binomial is a factor of the polynomial p(x).
If so, find the remaining factors of p(x).
p(x) = x³ + 2x² -x - 2 ; (x +2)
Answer:
a
Step-by-step explanation:
PLEASE HELP! To prepare for a bike race, Rex rides his bike for 12 miles each day for 3 days. The app he uses only tracks distance in kilometers. If 1 mile = 1.61 kilometers, what is Rex's distance in kilometers? Round the answer to the nearest hundredth. 7.45 kilometers 19.32 kilometers 22.36 kilometers 57.96 kilometers
Based on the distance that Rex rode every day for three days, Rex's distance in 3 days in kilometers can be found to be 57.96 kilometers.
How to find the distance in miles?First, find the distance that Rex rode in those three days in miles. This can be found as:
= Number of miles rode per day x Number of days
= 12 x 3
= 36 miles
Then convert this to kilometers.
If one mile is 1.61 kilometers, then 36 miles would be:
= Number of miles x Miles per kilometer
= 36 x 1.61
= 57.96 kilometers
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If Rex rides his bike for 12 miles each day for 3 days. Then distance in kilometers is 57.96.
What is Distance?The length along a line or line segment between two points on the line or line segment.
Speed=Distance / Time
Distance=Speed × Time.
Given that Rex rides his bike for 12 miles each day for 3 days
and 1 mile = 1.61 kilometre.
Let us convert 12 miles to kilometres
12×1.61=19.32 km
Now let us calculate the Distance as the speed is 19.32km and time is 3 days.
By the formula to get distance we have to multiply speed and time.
Distance=19.32×3
=57.96
Hence Rex's distance in kilometers is 57.96.
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PLS HELP ASAP
Clara and Toby are telemarketers.
Yesterday, Clara reached 4 people in 10 phone calls, while Toby reached 3 people in 8 phone calls.
If they continue at those rates, who will reach more people in 40 phone calls?
Use the drop-down menu to show your answer.
Evaluate g(-3)Determine the coordinates of the point given by the answer aboveEvaluate g(2a)Step By Step Explanation Please
Given the quadratic equation:
[tex]g(x)=3x^2-5x+4[/tex]Let's solve for the following:
• (a) g(-3)
To solve for g(-3), substitute -3 for x and evaluate.
Thus, we have:
[tex]\begin{gathered} g(x)=3x^2-5x+4 \\ \\ g(-3)=3(-3)^2-5(-3)+4 \\ \\ g(-3)=3(9)+15+4 \\ \\ g(-3)=27+15+4 \\ \\ g(-3)=46 \end{gathered}[/tex]Hence, we have:
g(-3) = 46
• (b) To determine the coordinates of the point given in question (a).
In the function, g(x) can also be written as y.
Thus, from g(-3), we have the following:
x = -3
y = 46
When x = -3, the value of y = 46
In point form, we have the coordinates:
(x, y) ==> (-3, 46)
Therefore, the coordinates of the given point by the answer in (a) is:
(-3, 46)
• (c) Evaluate g(2a).
To evaluate g(2a), substitute 2a for x in the equation and evaluate.
Thus, we have:
[tex]\begin{gathered} g(x)=3x^2-5x+4 \\ \\ g(2a)=3(2a)^2-5(2a)+4 \\ \\ g(2a)=3(4a^2)-5(2a)+4 \\ \\ g(2a)=12a^2-10a+4 \end{gathered}[/tex]ANSWERS:
• (a) g(-3) = 46
• (b) (-3, 46)
• (c) g(2a) = 12a² - 10a + 4
You have to write 1/2 page for an assignment. You write 1/5 page. How many pages do you have left to write ?
To find the number of missing pages:
[tex]\frac{1}{2}-\frac{1}{5}=[/tex]rewriting the expression as homogeneous fractions:
[tex]\frac{1}{2}\times\frac{5}{5}-\frac{1}{5}\times\frac{2}{2}=[/tex]simplifying it:
[tex]\frac{5}{10}-\frac{2}{10}=\frac{3}{10}[/tex]ANSWER
you have left 3/10 page.
I need help question 10 b and c
Part b.
In this case, we have the following function:
[tex]y=5(2.4)^x[/tex]First, we need to solve for x. Then, by applying natural logarithm to both sides, we have
[tex]\log y=\log (5(2.4^x))[/tex]By the properties of the logarithm, it yields
[tex]\log y=\log 5+x\log 2.4[/tex]By moving log5 to the left hand side, we have
[tex]\begin{gathered} \log y-\log 5=x\log 2.4 \\ \text{which is equivalent to} \\ \log (\frac{y}{5})=x\log 2.4 \end{gathered}[/tex]By moving log2.4 to the left hand side, we obtain
[tex]\begin{gathered} \frac{\log\frac{y}{5}}{\log2.4}=x \\ or\text{ equivalently,} \\ x=\frac{\log\frac{y}{5}}{\log2.4} \end{gathered}[/tex]Therfore, the answer is
[tex]f^{-1}(y)=\frac{\log\frac{y}{5}}{\log2.4}[/tex]Part C.
In this case, the given function is
[tex]y=\log _{10}(\frac{x}{17})[/tex]and we need to solve x. Then, by raising both side to the power 10, we have
[tex]\begin{gathered} 10^y=10^{\log _{10}(\frac{x}{17})} \\ \text{which gives} \\ 10^y=\frac{x}{17} \end{gathered}[/tex]By moving 17 to the left hand side, we get
[tex]\begin{gathered} 17\times10^y=x \\ or\text{ equivalently,} \\ x=17\times10^y \end{gathered}[/tex]Therefore, the answer is
[tex]f^{-1}(y)=17\times10^y[/tex]PLEASE HELP ME!! a shoe company is going to close one of its two stores and combine all the inventory from both stores these polynomials represented the inventory in each store. which expression represents the combined inventory of the two stories?
Add the two expressions together;
[tex]\begin{gathered} (\frac{1}{2}g^2+\frac{7}{2})+(3g^2-\frac{4}{5}g+\frac{1}{4}) \\ =\frac{1}{2}g^2+3g^2-\frac{4}{5}g+\frac{7}{2}+\frac{1}{4} \\ =3\frac{1}{2}g^2-\frac{4}{5}g+(\frac{14+1}{4}) \\ =\frac{7}{2}g^2-\frac{4}{5}g+\frac{15}{4} \end{gathered}[/tex]The first option is the correct answer
A pendulum swings through an angle of 14° each second. If the pendulum is 14 cm in length and the complete swing from right to left last two seconds what area is covered by each complete swing?
Answer;
[tex]\text{Area = 47.90 cm}^2[/tex]Explanation;
Firstly, we need a diagrammatic representation to get what is described in the question.
We have this as follows;
Now, from what we have here, the total angle swept by the pendulum moving from left to right is 28 degrees
To get the area, we simply need to find the area of the sector formed by the by pendulum
Mathematically, we have the area of a sector calculated as follows;
[tex]A\text{ = }\frac{\theta}{360}\times\pi\times R^2[/tex]theta is the angle made by the pendulum in one complete swing which is 28 degrees
pi is 22/7
R is the length of the pendulum which is 14 cm
Substituting these values in the formula above, we have it that;
[tex]\begin{gathered} A=\frac{28}{360}\times\frac{22}{7}\times14^2 \\ \\ A=47.90cm^2 \end{gathered}[/tex]Business Mathematics question
Number system depends on two basic concepts are Binary and Decimal.
Given the statement is :
Number System depends on two basic concept.
Let's know the definition of number system:
What is Number System?
A number system is defined as a system of writing to express numbers. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation of every number and represents the arithmetic and algebraic structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, multiplication and division.
Hence, Number system depends on two basic concepts are Binary and Decimal.
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Please help me with this problem just wanted to be sure that I am correct in order to help my son to under stand the break down of this problem. I believe that the answer is -3 but I am not sure please help?Solve for x.14x−1/2(4x+6)=3(x−4)−18 Enter your answer in the box.x =
SOLUTION
We want to solve for x in the equation
[tex]14x-\frac{1}{2}\mleft(4x+6\mright)=3\mleft(x-4\mright)-18[/tex]First we expand the brackets in both sides of the equation, this becomes
[tex]\begin{gathered} 14x-\frac{1}{2}(4x+6)=3(x-4)-18 \\ 14x-2x-3=3x-12-18 \end{gathered}[/tex]Note that the minus sign multiplies the items in the brackets too
Now, we collect like terms we have
[tex]\begin{gathered} 14x-2x-3x=-12-18+3 \\ 9x=-27 \\ \text{divide both sides by 9, we have } \\ \frac{9x}{9}=\frac{-27}{9} \\ x=-3 \end{gathered}[/tex]Hence x = -3
For Hox)=2x– 9 and 96 = ; « +9), find (10 g)(x) and (gof)(x). Then determine whether (f = 9/8)= (4 * H(X).What is (fog)x)?(10 g)x)=0
Given the functions;
[tex]\begin{gathered} f(x)=2x-9 \\ g(x)=\frac{1}{2}(x+9) \end{gathered}[/tex]We want to find the composite functions;
[tex]undefined[/tex]8. Factor ()=63−252++60 completely given that x=3 is a zero of p(x). Use only the techniques from the lecture on 3.3 (synthetic division). Other methods will receive a score of zero. Be sure to show all your work (including the synthetic division).
Factor the polynomial
[tex]\begin{gathered} p(x)=6x^3-25x^2+x+60 \\ \text{Given that, }x=3\text{ is a zero} \end{gathered}[/tex]Using the synthetic division method to factorize the polynomial completely,
The resulting coefficients from the table are 6, -7, -20, 0
Thus the quotient is
[tex]6x^2-7x-20[/tex]Factorizing the quotient completely,
[tex]\begin{gathered} 6x^2-7x-20 \\ =6x^2-15x+8x-20 \\ =3x(2x-5)+4(2x-5) \\ =(3x+4)(2x-5) \end{gathered}[/tex]Therefore, the other two zeros of the polynomial are:
[tex]\begin{gathered} (3x+4)(2x-5)=0 \\ 3x+4=0 \\ x=-\frac{4}{3} \\ 2x-5=0 \\ x=\frac{5}{2} \\ \\ Therefore,t\text{he factors of the polynomial are:} \\ (x-3)(3x+4)(2x-5) \end{gathered}[/tex]S = 2^0 + 2^1 + 2^2 + 2^3 + ...... 2^99a) Show that S can be divided by 15b) Show that S has at least 30 digits
Answer:
Explanation:
Here, we want to show that the sum is divided by 15
From what we have, the given sum is a geometric sequence
The first term is 1
Now, the pattern of ending afterwards will be 2, 4, 6 and 8
This ending keeps repeating itself
This 2,4,6,8 pattern could repeat itself 24 times
So we have a total of 1 + 24(4) = 97 terms
To make it 100, we have the last three terms as 2,4 and 8
So we have the ending number ONLY sum as follows:
1 + 24(2+4+6+8) + 2 + 4 + 8 = 1 + 480 + 14 = 495
We can divide this by 15 and that gives 495/15 = 33
That shows that the sum is divisible by 15
Secondly, we want to show that S has at least 30 digits
We can infer this from the last terms
We can write 2^99 as 2^33 * 2^33 * 2^33
A single 2^33 has a value of 8,589,934,592
That means 10 digits
The other two has 10 digits too
The sum of all possible digits in the largest term is 10 + 10 + 10 = 30
That makes a total of 30
The question states 30 or more
Hence, this is correct
find the area of the composite figures by either adding and subtracting regions
Explanation:
This figure is a rectangle and a quarter of a circle. We can find their areas and add them to find the total area of the figure.
The area of the rectangle is:
[tex]A_{\text{rectangle}}=17cm\times10\operatorname{cm}=170\operatorname{cm}^2[/tex]The area of a circle is:
[tex]A_{\text{circle}}=\pi\cdot r^2[/tex]Where r is the radius of the circle. In this case we have a quarter of a circle, so its area is a quarter of the area of the circle:
[tex]A_{1/4\text{circle}}=\frac{A_{\text{circle}}}{4}=\frac{\pi\cdot r^2}{4}[/tex]The radius of this circle is 8cm:
[tex]A_{1/4\text{circle}}=\frac{\pi\cdot8^2}{4}=\frac{\pi\cdot64}{4}=\pi\cdot16\approx50.27\operatorname{cm}^2[/tex]The total area of the figure is:
[tex]A_{\text{figure}}=A_{\text{rectangle}}+A_{1/4\text{circle}}=170\operatorname{cm}+50.27\operatorname{cm}=220.27\operatorname{cm}^2[/tex]Answer:
The area is 220.27 cm²
