Current ages
Jessica's age = x
Jessica's father = x + 24
In 3 years time, there ages will be:
Jessica's age = x+ 3
Jessica's father = x + 24 + 3 = x + 27
But Jessica's father will be 5 times her age
Hence;
x + 27 = 5(x+3)
Open the parenthesis
x + 27 = 5x + 15
collect like term
5x - x = 27 - 15
4x = 12
Divide both-side of the equation by 4
x = 3
In the current time;
Jessica is 3 years old
Jessica's father is x + 24 = 3 + 24 = 27 years old
The function graphed is of the form y=a sin bx or y=a cox bx, where b>0. Determine the equation of the graph.
The equation is of the form y=asinbx
The amplitude of the graph is a=3 unit
The period is
[tex]\frac{\pi}{2}[/tex]Thus,
[tex]\frac{2\pi}{b}=\frac{\pi}{2}\Rightarrow b=4[/tex]Thus the equation becomes,
y=3sin(4x)
A town's population grows at a rate proportional to itspopulation. If the growth rate is 6.8% per year and thecurrent population is 1627, what will the population be8.5 years from now?a) Write the equationb) Determine the future population of the town.
Solution:
Given:
[tex]\begin{gathered} P_o=1627 \\ r=6.8\text{ \%}=\frac{6.8}{100}=0.068 \end{gathered}[/tex]The equation of the population can be gotten using the formula;
[tex]P=P_oe^{rt}[/tex]Hence, the equation is:
[tex]P=1627e^{0.068t}[/tex]The population of the town 8.5years from now will be:
[tex]\begin{gathered} when\text{ }t=8.5years \\ \\ Then, \\ P=1627e^{0.068\times8.5} \\ P=1627e^{0.578} \\ P=2900.08 \\ \\ Since\text{ the population must be a whole number,} \\ P=2900 \end{gathered}[/tex]What is the product of -a + 3)(a + 4)?A. a 2-a + 12 B. a 2 - a - 12 C. -a 2 - a - 12 D. -a 2 - a + 12
The given expression is : (-a +3)(a + 4)
[tex]\begin{gathered} (-a+3)(a+4)=(-a)\times(a+4)+3(a+4)_{}_{} \\ (-a+3)(a+4)=(-a)\times(a)+4\times(-a)+3\times(a)+3\times4 \\ (-a+3)(a+4)=-a^2-4a+3a+12 \\ (-a+3)(a+4)=-a^2-a+12 \end{gathered}[/tex]Answer : D) - a² - a + 12
Find the area of each figure. 4. & 3 3 S 2 2 8 a square units
area is
[tex]A=s^2=1^2=1[/tex]and
[tex]\text{Atotal}=1\times11=11[/tex]answer: 11 square units
What Postulate or theorem proves that these triangles are similar?
Solution
The Side-Angle-Side (SAS) Theorem states if two sides of one triangle are proportional to two corresponding sides of another triangle, and their corresponding included angles are congruent, the two triangles are similar.
Next
AC is corresponding to EC
BC is corresponding to DC
[tex]\begin{gathered} \frac{AC}{EC}\text{ = }\frac{6}{12}\text{ = }\frac{1}{2} \\ \frac{BC}{DC}\text{ = }\frac{5}{10}\text{ = }\frac{1}{2} \end{gathered}[/tex]The ratio of their corresponding sides is proportional.
Final answer
The Side-Angle-Side (SAS) Theorem
convert 7π\12 from radians to degrees
ANSWER:
[tex]\frac{7}{12}\pi=105\text{\degree}[/tex]STEP-BY-STEP EXPLANATION:
To convert degrees to radians or radians to degrees, we must bear in mind that pi equals 180°, therefore:
[tex]\frac{7}{12}\pi\cdot\frac{180\text{\degree}}{\pi}=105\text{\degree}[/tex]The concert lasted two hours. For how many minutes did the chorus perform alone?
In order to find how many minutes the chorus performed, let's convert the time of 2 hours into minutes.
To do that conversion, we need to know that 1 hour is equal to 60 minutes.
Then, we can write the following rule of three:
[tex]\begin{gathered} \text{hours}\to\text{minutes} \\ 1\text{ hour}\to60\text{ minutes} \\ 2\text{ hours}\to x\text{ minutes} \end{gathered}[/tex]From this rule of three, we can write the following equation and solve it for x:
[tex]\begin{gathered} \frac{1}{2}=\frac{60}{x} \\ 1\cdot x=2\cdot60 \\ x=120 \end{gathered}[/tex]Therefore the chorus performed for 120 minutes.
Which one of the following graphs is the graph of:if x > 0If(x) = -1 if x = 0 ?-3if x < 0OA.GB.Ax
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Explain the given piecewise function
We look for where x is greater than zero and f(x) will be x
Where x equals 0 and the f(x) will be -1
Where x is less than 0 and f(x) will be -3
Hence, the correct graph will be:
OPTION D
Social Security numbers are of the formXYZ - MN - ABCD.The first three digits XYZ are referred toas the area number. The number isassociated with where you registered foryour social security card. X is a number 1- 7. Y and Z can be any number 0 – 9.Also the area code 666 has never beenused. The middle two digits MN arereferred to as the group number. Thegroup number can be any number from01 to 99 (it cannot be 00). The last 4 digitsABCD are the serial number. The serialnumber can be any from 0001 to 9999 (itcannot be 0000). Additionally, thenumber social security number 123-45-6789 is only used in advertisements. Howmany social security numbers arepossible?
Solution
For XYZ
[tex]\begin{gathered} (7\times10\times10)\text{ = 700} \\ 700-1=699\text{ways} \\ \end{gathered}[/tex]For MN
[tex]\begin{gathered} (10\times10)\text{ =100} \\ 100-1=99\text{ways} \end{gathered}[/tex]For ABCD
[tex]\begin{gathered} (10\times10\times10)\text{ =10,000} \\ 10,000-1\text{ =9,999ways} \end{gathered}[/tex]Finally
[tex]\begin{gathered} (699\times99\times9999)-1 \\ =691,940,798\text{ } \end{gathered}[/tex]Social security numbers are
possible 691, 940,798
is the number 6.35 a natural number
The number is 6.35.
The Natural numbers are all numbers 1, 2, 3, 4… They are the numbers you usually count and they will continue on into infinity.
6.35 is a natural number.
True or False?Dependent is all events which are not included in the specified event.
Two events are independent, statistically independent if the occurrence of one does not affect the probability of occurrence of the other. So the answer is false.
A psychology test has personality questions numbered 1,2,3, intelligence questions numbered 1,2,3,4, and attitude questions numbered 1,2. If a single question is picked at random, what is the probability that the question is an intelligence question OR has an odd number?Provide the final answer as a simplified fraction.
Given:
Personality question = 1,2,3
Intelligence question = 1,2,3,4
Attitude question = 1,2
Find-: what is the probability that the question is an intelligence question OR has an odd number
Sol:
The total number of questions is:
3 Personality, 4 intelligence, and 2 attitudes so the total number of questions is:
[tex]\begin{gathered} =3+4+2 \\ \\ =9 \end{gathered}[/tex]The odd numbers are:
[tex]\begin{gathered} =1,1,1,3,3 \\ \\ \text{ Total 5 questions.} \end{gathered}[/tex]Probability is :
[tex]P=\frac{\text{ Favorable outcome }}{\text{ Total outcome}}[/tex]The probability of an odd number is:
[tex]P=\frac{5}{9}[/tex]The probability of intelligence question of even number is:
[tex]P=\frac{2}{9}[/tex]So required probability is:
[tex]\begin{gathered} =\frac{5}{9}+\frac{2}{9} \\ \\ =\frac{7}{9} \end{gathered}[/tex]Probability is 7/9.
Sam is making Apple Pies and Pumpkin Pies for her Pie shop. She sells eachApple Pie for $6 and each Pumpkin Pie for $7. If Sam sells a total of 80 piesone day and makes a total of $512, how many apple pies did she sell?
Let m represent the number of apple pies Sam is making and
Let n represent the number of pumpkin pies Sam is making
If Sam sells a total of 80 pies, this can be represented by
m + n = 80 ------------------------------equation (1)
If she makes a total of $512 on the same day, this cost can be represented by
6m + 7n = 512 ----------------------------equation (2)
Solving both equations simultaneously
m + n = 80
6m + 7n = 512
from m + n = 80,
m = 80 - n
substitute m = 80 - n into equation (2)
we have
6(80 -n) + 7n = 512
480 - 6n + 7n = 512
480 + n = 512
n = 512 - 480
n = 32
Put n = 32 into m = 80 - n
we have
m = 80 - 32
m = 48
The number of apple pies Sam sell is 48
Richard and Teo have a combined age of 32. Richard is 5 years older than twice Teo's age. How old are Richard and Teo?
The age of Teo is 9 years and the Age of Richard is 23 years.
What is problem-solving?
Problem-solving is the act of defining a problem; figuring out the purpose of the trouble; identifying, prioritizing, and selecting alternatives for an answer; and imposing an answer.
Problem-solving starts with identifying the issue. As an example, a trainer may need to parent out a way to improve a scholar's overall performance on a writing talent test. To do that, the instructor will overview the writing exams looking for regions for improvement.
Calculation:-
Richard and Teo have a combined age = of 32
Richard is 7 years older than twice Teo's age
Let the age of Teo = X
Age of Richard = 2X + 7
2X + 5+ X = 31
X = 31 - 5 /3
X = 27/3
= 9 years
Therefore the age of Teo is 8 years
Age of Richard = 19 years
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what value of X makes this equation true?10×-6=3[×+1/2]A=14/15B=13/14C=15/14D=14/13
10x - 6 = 3 ( x + 1/2)
To find the value of x that make the equation true, we need to solve for x
open the parenthesis
10x - 6 = 3x + 3/2
collect like term
10x - 3x = 3/2 + 6
[tex]7x\text{ =}\frac{3+12}{2}[/tex][tex]7x\text{ =}\frac{15}{2}[/tex]Multiply both-side of the equation by 1/7
[tex]x\text{ =}\frac{15}{2}\times\frac{1}{7}[/tex][tex]x=\frac{15}{14}[/tex]Answer:
x=15/14
Step-by-step explanation:
No explanation
In the figure below, AC || DE, BD measures 8 m, AD measures 12 m, and BE measures 6 m. Find the length of BC.
Answer:
BC = 15 units===============
Parallel lines divide the transversal into proportional segments.
Set ratios and find the missing lengthBD/AD = BE/ED8/12 = 6/ED2/3 = 6 / EDED = 6*3/2ED = 9Find BCBC = BE + EDBC = 6 + 9BC = 15The first equation in a system is 5x+2y=-4Which equation gives a system with no solution
System of equations
A system of equations with two variables can have one solution, no solution, or infinitely many solutions.
Each equation corresponds to a line which can be expressed like:
y = mx + b
Where m is the slope and b is the y-intercept
For a system to have one solution, both lines must have different slopes, so they cross each other at one point.
If a system has no solution, both lines are parallel
If a system has infinitely many solutions, both lines are the same line (they coincide)
We have the following equation:
5x + 2y = -4
Let's solve it for y:
2y = - 5x -4
[tex]y=-\frac{5}{2}x-2[/tex]Only one of the lines has the same slope as the given line:
[tex]y=-\frac{5}{2}x-3[/tex]Both lines have the same slope but don't have the same y-intercept, so they are parallel.
Thus, the correct choice is D.
Answer:d
Step-by-step explanation:
Dawn's suitcase weighed 25 kilograms. Convert this weight to pounds. (Round youranswer to the nearest tenth.)
Given: Dawn's suitcase weighed 25 kilograms.
Required: To convert the given weight to pounds.
Explanation: Since 1 kilogram is equivalent to 2.20462 pounds. Hence to convert 25 kg to pounds, we need to multiply by 2.20462.
Thus we have
[tex]\begin{gathered} 25\text{ kg}=25\times2.20462\text{ pounds} \\ =55.1156\text{ pounds} \\ \approx55.1\text{ pounds} \end{gathered}[/tex]Final Answer: Dawn's suitcase weighed 55.1 pounds.
Parker is playing a game in which the object is to obtain a 0 by adding the points scored during each round. (Graph up top ) Based on the points he scored during the first 4 rounds, what score does Parker need on the 5th round?
Objective:
Obtain an score of 0 after add all the scores in all the rounds
Info given:
Round 1: -9
Round 2: 3
Round 3: -8
Round 4: 2
And we want to estimate the score in round 5 in order to obtain a 0 so we can do this:
[tex]-9+3-8+2+x=0[/tex]Where x represent the score required for round 5 and solving we got:
[tex]x=9+8-3-2=12[/tex]So then the score required for round 5 is 12
2) The debate team with 18 members is to choose four officers: captain, co-captain, treasurer, and secretary. How many ways can those officers be selected?
We have 18 members that can be positioned in 4 positions.
As the positions are different from each other, the order matters.
Then, this is a permutation of 18 elements in 4 places with no repetition.
The number of permutations can be calculated as:
[tex]\begin{gathered} P(n,r)=\frac{n!}{(n-r)!} \\ P(18,4)=\frac{18!}{(18-4)!}=\frac{18!}{14!}=18\cdot17\cdot16\cdot15=73440 \end{gathered}[/tex]We could have derived this by doing this analysis:
We have 18 to choose for captain.
Then, we have 17 left to choose for co-captain.
Finally 16 can be chosen for treasurer and 15 are left for secretary.
Then, the posible teams are 18*17*16*15=73440.
Answer: we can select them in 73,440 ways.
Write an equation of a circle giving its Center and radius or diameter.
The equation of a circle with center (h,k) and radius r is given as:
[tex](x-h)^2+(y-k)^2=r^2[/tex]From the question, it is given that the center of the circle is (-7,-4) and the diameter is 8.
Since the radius is half the diameter, it follows that the radius is:
[tex]\frac{\text{diameter}}{2}=\frac{8}{2}=4[/tex]So it implies that r=4, h=-7, and k=-4.
Substitute these values into the equation of a circle:
[tex]\begin{gathered} (x-(-7))^2+(y-(-4))^2=4^2 \\ \Rightarrow(x+7)^2+(y+4)^2=16 \end{gathered}[/tex]Hence, the equation of the circle is:
[tex](x+7)^2+(y+4)^2=16[/tex]
(4k² + 9k + 9) - (k² + 8k + 5)
we have the expression
(4k² + 9k + 9) - (k² + 8k + 5)
simplify
step 1
group like terms
(4k²-k²) +(9k-8k)+(9-5)
step 2
combine like terms
3k²+k+4Find the equation of a line perpendicular to the given line and pass through the given point. Write equation in slope-intercept form Line y= -4/5x + 2 point, (8,9) Graph
ANSWER
Equation:
[tex]y=\frac{5}{4}x-1[/tex]Graph:
The red line is the graph of the given line and the green line is the graph of the perpendicular line passing through (8, 9).
EXPLANATION
The equation of the line is given in the slope-intercept form,
[tex]y=-\frac{4}{5}x+2[/tex]The slope is -4/5 and the y-intercept is 2.
Two lines are perpendicular if their slopes are opposite reciprocals of each other. Therefore, a perpendicular line to the given line has a slope of 5/4,
[tex]y=\frac{5}{4}x+b[/tex]There is an infinite number of perpendicular lines, but there is only one that passes through the point (8, 9). Replace x and y with the coordinates of this point in the equation above,
[tex]9=\frac{5}{4}\cdot8+b[/tex]And solve for b,
[tex]\begin{gathered} 9=5\cdot2+b \\ 9=10+b \\ 9-10=b \\ -1=b \end{gathered}[/tex]Hence, the equation of the perpendicular line is,
[tex]y=\frac{5}{4}x-1[/tex]Katy has $40 in a savings account. The interest rate is 5%, compounded annually.To the nearest cent, how much interest will she earn in 3 years?What’s the answer
Given:
Katy has $40 in a savings account.
So, the principal = P = 40
The interest rate = r = 5% = 0.05
compounded annually, n = 1
Time = t = 3 years
So, Amount of money after 3 years = A
[tex]\begin{gathered} A=P\cdot(1+\frac{r}{n})^{nt} \\ A=40\cdot(1+\frac{0.05}{1})^3=46.305 \end{gathered}[/tex]So, the interest = A - P =
[tex]46.305-40=6.305[/tex]Rounding to the nearest cent
So, the answer will be:
interest she will earn in 3 years = $6.3
:) 4 inches of snow in 5 hoursHow much snow fell each hour?
The amount of snow fell in 5 hours is 4 inches.
To find snow fell in each hour, we divide 4 by 5.
Divide 4 by 5.
[tex]\frac{4}{5}=0.8[/tex]So 0.8 inches of snow fell each hour.
Answer: 0.8 inches
please help me ASAP!!!!
The phrase can be written as:
[tex]\lvert x+5\rvert=3[/tex]therefore the correct choice is the third one.
10. You and your date go to a restaurant where there are 5 meats, 6 vegetables, 4 types of bread, and 3 desserts to choose from. In how many ways could you select 2 meats, 3 vegetables, 1 bread, and 2 dessertsformula : nCr= n! / (nr) r!
You have to select 2 meats from 5 possible meats, that is, 5C2 =
[tex]5C2=\frac{5!}{(5-2)!\cdot2!}=\frac{120}{6\cdot2}=\frac{120}{12}=10[/tex]You have to select 3 vegetables from 6 possible vegetables, that is, 6C3 =
[tex]6C3=\frac{6!}{(6-3)!\cdot3!}=\frac{720}{6\cdot6}=\frac{720}{36}=20[/tex]You have to select 1 bread from 4 possible types of bread, that is, 4C1 =
[tex]4C1=\frac{4!}{(4-1)!\cdot1!}=\frac{24}{6\cdot1}=4[/tex]You have to select 2 desserts from 3 possible desserts, that is, 3C2 =
[tex]3C2=\frac{3!}{(3-2)!\cdot2!}=\frac{6}{1\cdot2}=3[/tex]The total possibilities are:
5C2*6C3*4C1*3C2 = 10*20*4*3 = 2400
What is the rule for the following dilation? A(-5.8) B(13.65, 12.8) C(-9, -13) D(0, 12) A'(-2.5, 4) B'(6.825, 6.4) C'(-4.5, -6.5) D'(0, 6) O Scale Factor: 1.5 O Scale Factor: 2 O Scale Factor: 5 O Scale Factor: 0.5
Answer:
(D)Scale Factor: 0.5
Explanation:
The coordinates A,B, C and D are those of the pre-image while A',B',C' and D' are those of the image.
Using the coordinates of A and A'
Let the scale factor = k
On the x-axis:
[tex]\begin{gathered} -5k=-2.5 \\ k=\frac{-2.5}{-5}=0.5 \end{gathered}[/tex]Similarly, on the y-axis
[tex]\begin{gathered} 8k=4 \\ k=\frac{4}{8} \\ k=0.5 \end{gathered}[/tex]We conclude that the scale factor is 0.5
hich is equivalent to RootIndex 5 StartRoot 1,215 EndRoot Superscript x?
243x
1,215 Superscript one-fifth x
1,215 Superscript StartFraction 1 Over 5 x EndFraction
243 Superscript StartFraction 1 Over x EndFraction
The given expression is equivalent to [tex]$(1215)^{x/5}[/tex].
What is a expression? What is a mathematical equation? What is Equation Modelling?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have the following equation -
[tex]$(\sqrt[5]{1215})^{x}[/tex]
For [tex]$\sqrt[a]{x} = x^{1/a}[/tex]
Using the rule, we can write -
[tex]$(\sqrt[5]{1215})^{x}[/tex] = [tex]$(1215)^{x/5}[/tex]
Therefore, the given expression is equivalent to [tex]$(1215)^{x/5}[/tex].
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Find the x- and y- intercept of each linear equation.1/2x + 1/4y = 3/2
hello
the equation given is
[tex]\frac{1}{2}x+\frac{1}{4}y=\frac{3}{2}[/tex]let's rewrite the equation
[tex]\begin{gathered} \frac{1}{2}x+\frac{1}{4}y=\frac{3}{2} \\ \frac{1}{4}y=\frac{3}{2}-\frac{1}{2}x \\ \text{ multiply through by 4} \\ \frac{4}{4}y=4\times\frac{3}{2}-(4\times\frac{1}{2}x) \\ y=\frac{12}{2}-2x \\ y=6-2x \end{gathered}[/tex]now let's find the x and y intercept now
to do this, put x = 0 and solve and then put y = 0 and then solve
[tex]\begin{gathered} y=6-2x \\ \text{put x = 0} \\ y=6-2(0) \\ y=6-0 \\ y=6 \end{gathered}[/tex]now let's put y = 0
[tex]\begin{gathered} y=6-2x \\ 0=6-2x \\ 2x=6 \\ \text{divide both sides by 2} \\ \frac{2x}{2}=\frac{6}{2} \\ x=3 \end{gathered}[/tex]now the coordinates for the equation is (3, 6)