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The intercept between both lines, represents the solution to the system compounded by the equations for both lines. It means (3,4) is the solution to both lines A and B.
Related Questions
10x2+4x factor completely
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Answer:
2x * ( 5x + 2 )
Step-by-step explanation:
10x^2 = 2x * 5x
4x = 2x * 2
10x^2 + 4x = 2x * ( 5x + 2 )
Based on the triangles shown below, which statements are true? Select All that apply.
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Answer:
All the options except the third choice are correct.
Explanation:
In the given figure:
[tex]\angle\text{GER}\cong\angle\text{TEA (Vertical Angles)}[/tex]Since angles G and T are congruent:
• Triangles GER and TEA are similar triangles.
Therefore, the following holds:
[tex]\begin{gathered} \triangle\text{GRE}\sim\triangle\text{TAE} \\ \triangle E\text{GR}\sim\triangle E\text{TA} \\ \frac{GR}{TA}=\frac{RE}{AE} \end{gathered}[/tex]Similarly:
[tex]\begin{gathered} \frac{EG}{ET}=\frac{GR}{TA} \\ ET=10,EG=5,TA=12,RG=\text{?} \\ \frac{5}{10}=\frac{RG}{12} \\ \frac{1}{2}=\frac{RG}{12} \\ 2RG=12 \\ RG=\frac{12}{2} \\ RG=6 \\ \text{Therefore if }ET=10,EG=5,and\; TA=12,then\; RG=6 \end{gathered}[/tex]Finally, angles R and A are congruent.
[tex]\begin{gathered} m\angle R=m\angle A \\ 80\degree=(x+20)\degree \\ x=80\degree-20\degree \\ x=60\degree \end{gathered}[/tex]The correct choices are:
[tex]\begin{gathered} \triangle\text{GRE}\sim\triangle\text{TAE} \\ \triangle E\text{GR}\sim\triangle E\text{TA} \\ \frac{GR}{TA}=\frac{RE}{AE} \\ I\text{f }ET=10,EG=5,and\; TA=12,then\; RG=6 \\ \text{If }m\angle R=80\degree\text{ and }m\angle A=(x+20)\degree,then\; x=60\text{ } \end{gathered}[/tex]Only the third choice is Incorrect.
Given the kite ABCD, which statement is false?Just the answer.
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Explanation
let's check every option
then
A)
[tex]\angle ADC\text{ is congruente to }\angle ABC[/tex]we can see those angles (in black), aand as the lengths of the sides are similar this angles are congruente,so
[tex]\begin{gathered} \angle ADC\text{ is congruent to }\angle ABC \\ \text{true} \end{gathered}[/tex]B)
[tex]undefined[/tex]Sorry if it's a little blurryAlso this worksheet is about simplify
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Write four different equation with -3 as solution.
5x + 20 = 5 subtract 20 both sides
5x = 5 - 20
5x = -15 divide by 5 both sides
x = -15/5
x = -3
3x - 6 = -3x - 24 add 3x both sides
3x + 3x - 6 = - 24
6x - 6 = -24 add 6 both sides
6x = -24 + 6
6x = -18 divide by 6 both sides
x = -18/6
x = -3
1/3 x + 10 = 9 subtract 10 both sides
1/3 x = 9 -10
1/3 x = - 1 multiply by 3 both sides
x = -1(3)
x = -3
x + 23 = 20 subtract 23 both sides
x = 20 - 23
x = -3
In all previous procedures you constructed the equation by taking into account that x=-3, that is the key to determine the expressions left side and right side. For example in the last procedure for x + 23 = 20, you know that -3 plus 20 is equal to 20.
If the sum of a number and nine is tripled , the result is two less than twice the number. Find the number.
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By solving a simple linear equation we will see that the number is -29.
How to find the number?Let's define x as the number, then the sentence:
"If the sum of a number and nine is tripled , the result is two less than twice the number"
Can be written as the equation:
3*(x + 9) = 2x - 2
This is a linear equation that we can solve for x:
3*(x + 9) = 2x - 2
3x + 27 = 2x - 2
3x - 2x = -2 - 27
x = -29
The number is -29.
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Convert 185 pounds into kilograms
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We want to convert 185 pounds to kilograms
1 pound = 0.4536 kilograms
Therefore,
185 pounds = 185 * 0.4536
185 pounds = 83.916 kilograms
185 pounds is 83.916 kilograms
Eduardo has 45 yards of rupe light.Exactly how many more yards does he need to finish the car's ceiling?ItemLength of Rope Light (yds)Front spoilerRear spoilerCeilingDONENaimmisDashboardWhat is the fraction
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please explain What is the simplified form of the expression?
2x 2 + 4y + 3x 2 – 2y + 3y
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The simplified form of the expression is found to be 5(x² + y) by adding or subtracting all the similar terms
What is the difference between a mathematical expression and an equation?A number, a variable, or a mix of numbers, variables, and operation symbols make up an expression. Two expressions joined by an equal sign form an equation.
What does "simplification of an algebraic expression" mean?The technique of expressing an algebraic expression in the most effective and compact form without altering the original expression's value is known as simplification. The procedure involves gathering related terms, which calls for adding or removing terms from an expression.
The given expression is 2x² + 4y + 3x² -2y +3y
We need to simplify this expression.
2x² + 4y + 3x² -2y +3y
= 2x² + 3x² + 4y - 2y + 3y
= 5x² + 5y
= 5(x² + y)
Therefore, the simplified form of the expression is found to be 5(x² + y) by adding or subtracting all the similar terms.
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Triangle Ris a right triangle. Can we use two copies of TriangleR to compose a parallelogram that is not a square? Explain yourreasoningR.R
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If we have a right triangle R as:
We have a right angle.
The only way we can make a parallelogram that is not a square is placing the triangle so the parallelogram does not have any right angle. That would be;
This way, the right angles are added to one of the other angles, in order to have none right angles, as it is the condition to have a four-side figure that is not a square or rectangle.
(If the triangle R can compose a square, it should have two equal sides, not like our figure).
at one point during the summer, Marsha has read 500 pages of her summer reading assignment, and Jan has read read 460 pages. marsha reads reads 20 pages per week for the reminder of the summer, how many weeks,w,will it take before the girls have read the same number of pages?
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Answer:
500 + 20w = 460 + 30w
Explanation:
We will calculate an equation for the number of pages read by each girl.
Marsha has read 500 pages and she reads 20 pages per week. It means that after w weeks, she will read 20 times w plus the 500 initial pages, so:
M = 20w + 500
In the same way, Jan has read 460 pages and she reads 30 pages per week. So, the equation that model the number of pages that she reads after w weeks is:
J = 30w + 460
Now, we need to find w such that M and J would be equal, so, we will formulate the following equation:
M = J
20w + 500 = 30w + 460
500 + 20w = 460 + 30w
Therefore, the answer is:
500 + 20w = 460 + 30w
Given the venn diagram below, what is the correct notation!A. G∩(M∪F)′B. (M∩F)′C. none of theseD. (M∪F)′
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Sets
The image shows a Venn diagram of two sets M and F inside of a containing set G.
We must recall the following notations:
U = Union of sets. Everything inside of any of the sets.
∩ = Intersection between two sets. The common part of both sets.
' = Negation. Everything outside of a set or a set operation.
It's required to express in set notation the grayed part of the diagram. Note it's inside of G and outside of both M and F.
As stated above, the notation to express the union of sets is U and it's precisely the white region of the diagram. So M U F is the white part.
But we want the outside of that white part, so we use (M U F)'.
That is outside of the union, but it includes everything (the entire universe). So we must intercept that with G to take only the grayed part, so the answer is:
A. G∩(M∪F)′
Answer:
Step-by-step explanation:
The image shows a Venn diagram of two sets M and F inside of a containing set G.
We must recall the following notations:
U = Union of sets. Everything inside of any of the sets.
∩ = Intersection between two sets. The common part of both sets.
' = Negation. Everything outside of a set or a set operation.
It's required to express in set notation the grayed part of the diagram. Note it's inside of G and outside of both M and F.
As stated above, the notation to express the union of sets is U and it's precisely the white region of the diagram. So M U F is the white part.
But we want the outside of that white part, so we use (M U F)'.
That is outside of the union, but it includes everything (the entire universe). So we must intercept that with G to take only the grayed part, so the answer is:
A. G∩(M∪F)′
Kiran is flying a kite. He gets tired, so he stakes the kite into the ground. The kite is on a stringthat is 18 feet long and makes a 30 degree angle with the ground. How high is the kite?
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ANSWER:
9 feet
STEP-BY-STEP EXPLANATION:
We can calculate the value of the height of the kite by means of the trigonometric function sine, which is the following:
[tex]\begin{gathered} \sin \theta=\frac{\text{opposite}}{\text{hypotenuse}} \\ \theta=30\text{\degree} \\ \text{hypotenuse = 18} \\ \text{opposite }=x \end{gathered}[/tex]Replacing and solving for x:
[tex]\begin{gathered} \sin 30=\frac{x}{18} \\ x=\sin 30\cdot18 \\ x=9 \end{gathered}[/tex]The height of the kite is 9 feet
6. Cesium-137 has a half-life of 30 years. Suppose a lab stores 30 mg in 1975. How much would be left in 2065? y = a (1 + r) (Fill in answer choices for a, r and t.)
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The formula for calculating the amount remaining after a number of half years , n is :
[tex]\begin{gathered} A=\frac{A_{\circ}}{2^n^{}} \\ \text{where A}_{\circ}\text{ =initial }amount \\ n=\frac{t}{t_{\frac{1}{2}}} \end{gathered}[/tex]The lab store mass of Cesium-137 is 30mg in 1975
then the mass of Cesium-137 in 2065,
Time period =2065-1975
time period t=90 years,
substitute the value and solve for A
[tex]\begin{gathered} A=\frac{30}{2^{\frac{90}{45}}} \\ A=\frac{30}{2^2} \\ A=\frac{30}{4} \\ A=7.5\text{ mg} \end{gathered}[/tex]In 2065, the mass of Cesium -137 will be 7.5 mg
Answer : 7.5mg
Can you please help me
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length of arc PQ = 3.14 meters (option B)
Explanation:[tex]\begin{gathered} \text{Length of an arc in radians = r}\theta \\ \end{gathered}[/tex]The angle given is in degrees:
[tex]\text{length of an arc using angle in degr}ees\text{ = }\theta/360\times2\pi r[/tex][tex]\begin{gathered} \theta\text{ = 60}\degree \\ PR\text{ = radius =3m } \\ \text{let }\pi\text{ = 3.14} \\ \text{length of arc PQ = }\frac{60}{360}\times2\times3.14\times3 \end{gathered}[/tex][tex]\begin{gathered} \text{length of arc PQ = }\frac{1}{6}\times3.14\times6\text{ = 3.14} \\ \text{length of arc PQ = 3.14 meters ( option B)} \end{gathered}[/tex]which of the following could be the combination of values for the students and the minimum numbers of chaperones the museum requires
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3 chaperones ---------------------------- 24 students
9 chaperones --------------------------- 72 students
2chaprones 16 students
7.5 chaperones ----------------------- 60 students
5.6 chaperones------------------------ 45 students
5 chaperones ------------------------- 40
The first two options are correct
What is the 15th term in the sequence using the given formula?
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Solution:
The formula is given below as
[tex]c_n=3n-1[/tex]Concept:
To figure out the 15th term, we will substitute n=15
By substituting values, we will have
[tex]\begin{gathered} c_{n}=3n-1 \\ c_{15}=3(15)-1 \\ c_{15}=45-1 \\ c_{15}=44 \end{gathered}[/tex]Hence,
The final answer is
[tex]\Rightarrow44[/tex]The THIRD OPTION is the right answer
C. = 3n - 1
Concept:
To figure out the 15th term, we will substitute n=15
By substituting values, we will have
Cn = 3n - 1
C15 = 3(15) -1
C15= 45 - 1
C15 = 44
The answer is
44
7)Find the equation of the line that goes throughthe points (-1, 4) and (0, 5).Find m:Which point is the y-intercept?x43bEquation in the form y = mx + b:Graph the line:Y
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Given the following question:
Point A = (-1, 4) = (x1, y1)
Point B = (0, 5) = (x2, y2)
[tex]m=\frac{y2-y1}{x2-x1}[/tex][tex]\begin{gathered} m=\frac{y2-y1}{x2-x1} \\ m=\frac{5-4}{0--1}=\frac{1}{1}=1 \\ m=1 \end{gathered}[/tex]Now write in slope intercept form where...
[tex]\begin{gathered} y=mx+b \\ m=1 \\ y=4 \\ x=-1 \end{gathered}[/tex]Substitute to find b:
[tex]\begin{gathered} 4=1(-1)+b \\ 1\times-1=-1 \\ \text{ +1 on both sides} \\ 4+1=5 \\ 5=b \\ b=5 \\ y=1x+5 \end{gathered}[/tex]b = 5
y intercept is (0, 5)
Now graph the following equation:
please help me the blue line is what I have to find
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Weare given to complete a blank in the equation:
2 m - 3 - _____ + 17 = 14
So, we start by combining all the terms we can combine (the pure numerical terms that don't have the variable "m")
2 m - 3 + 17 - ______ = 14
2 m + 14 - _______ = 14
Now we subtract 14 from both sides of the equal sign:
2 m - ____ = 0
which means that the blank should be exactly "2m" such that subtracted from 2m gives a perfect zero.
Answer: complete the blank with "2 m"
-Quadratic Equations- Determine the number and the nature of the solutions to (3a + 24)² = -36 and then solve
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ANSWER
There are two solutions and they are both complex solutions. The solutions are:
[tex]a=2i-8;a=-2i-8[/tex]EXPLANATION
We want to determine the number and nature of solutions to the equation:
[tex](3a+24)^2=-36[/tex]To do this, solve the equation by first, finding the square root of both sides of the equation:
[tex]\begin{gathered} \sqrt[]{(3a+24)^2}=\pm\sqrt[]{-36}=\pm\sqrt[]{-1\cdot36} \\ \Rightarrow3a+24=\pm\mleft\lbrace\sqrt[]{36}\cdot\sqrt[]{-1}\mright\rbrace \\ 3a+24=\pm6i \end{gathered}[/tex]Now, solve the equation for a:
[tex]\begin{gathered} 3a=\pm6i-24 \\ \Rightarrow a=\pm\frac{6i}{3}-\frac{24}{3} \\ \Rightarrow a=2i-8;a=-2i-8 \end{gathered}[/tex]Hence, there are two solutions and they are complex solutions.
Write the first five terms of each sequence a(1) = 7, a(n) = a(n - 1) - 3 for n = 2.
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Answer:
7 , 4, 1, -2 and -5
Explanation:
Given a sequence such that:
[tex]\begin{gathered} a(1)=7 \\ a(n)=a(n-1)-3,n\geqslant2 \end{gathered}[/tex][tex]\begin{gathered} a\left(2\right)=a\left(2-1\right)-3=a(1)-3=7-3=4\implies a(2)=4 \\ a\left(3\right)=a\left(3-1\right)-3=a(2)-3=4-3=1\implies a(3)=1 \\ a\left(4\right)=a\left(4-1\right)-3=a(3)-3=1-3=-2\implies a(4)=-2 \\ a\left(5\right)=a\left(5-1\right)-3=a(4)-3=-2-3=-5\implies a(5)=-5 \end{gathered}[/tex]Therefore, the first five terms of the sequence are:
7 , 4, 1, -2 and -5
what is the range of the function graphed below?[tex]1 \leqslant y \ \textless \ 4 \\ - 3 \ \textless \ y \leqslant 3 \\ - 2 \leqslant y \leqslant 3 \\ - 3 \leqslant y \ \textless \ 4[/tex]
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The range of the function is (-3, 3]
The range of a function is composed by all the values that y reaches in the function. Here we can see that the functions goes from 3 to -3. Then the range set is (-3, 3]. It has a parentheses in -3 because the function doesn't reach -3
could someone help me find the measures of this Rhombus? im very confused right now and need an explanation on thisThe measures you need to find:NK=NL=ML=JM=M
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We shall take a quick reminder of the properties of a rhombus.
All sides are equal in measure
The opposite sides are parallel
The diagonals bisect each other at right angles
Opposite angles are equal in measure
Therefore, we can deduce the following from the given rhombus;
If JL bisects MK, then
[tex]\begin{gathered} MN=NK=\frac{MK}{2} \\ MN=NK=\frac{24}{2} \\ MN=NK=12 \end{gathered}[/tex]If MK bisects JL, then line
[tex]\begin{gathered} JN=NL=\frac{JL}{2} \\ JN=NL=\frac{20}{2} \\ JN=NL=10 \end{gathered}[/tex]Also, in triangle MJN,
MN = 12,
JN = 10,
Angle J = 50
Angle N = 90
Therefore angle M = 40
(All three angles in a triangle sum up to 180)
Therefore, in right angled triangle MJN, with the right angle at N,
[tex]\begin{gathered} MN^2+JN^2=JM^2 \\ 12^2+10^2=JM^2 \\ 144+100=JM^2 \\ 244=JM^2 \\ \sqrt[]{244}=JM \\ JM=15.6 \end{gathered}[/tex]All sides are equal, therefore,
JM = ML = 15.6
Since line MK has been bisected by line JL, then
[tex]\angle KNL=90[/tex]Also angle MJL equals 50, and line JL bisects angle J, then
[tex]\angle MJL=\angle KJL=50[/tex]If angle MJL and angle KJL both measure 50, then angle MJK equals 100 (50 + 50).
Opposite angles of a rhombus are equal, hence
[tex]\angle MJK=\angle MLK=100[/tex]If KJL = 50, and JNK = 90, then
[tex]\begin{gathered} \angle JKM+\angle KJL+\angle JNK=180\text{ (angles in a triangle sum up to 180)} \\ \angle JKM+50+90=180 \\ \angle JKM=180-50-90 \\ \angle JKM=40 \end{gathered}[/tex]If JKM = 40, then
[tex]\begin{gathered} \angle JKM=\angle LKM=40 \\ \angle JKL=\angle JKM+\angle LKM \\ \angle JKL=80 \\ \angle JKL\text{ and }\angle JML\text{ are opposite angles. Therefore,} \\ \angle JML=80 \end{gathered}[/tex]So the answers are;
NK = 12
NL = 10
ML = 15.6
JM = 15.6
Solve each equation mentally. 2×=10. -3×=21
Answers
Explanation:
To solve this you have to divide the number on the right side by the coefficient of x:
2x=10 --> 10/2=5
-3x=21 --> 21/(-3)=-7
Answers:
2x=10 --> 5
21/(-3) --> -7
Amanda likes to launch model rockets. For one of Amanda's rockets, the function S(t)= −16t^2+41t+112 gives the height of the rocket above the ground in feet, in terms of the number of seconds t since the rocket's engine stops firing.Please use 4 or more decimals.How far above the ground is the rocket when it stops firing?After how many seconds does the rocket reach its maximum height?What is the maximum height reached by the rocket?After how many seconds will the rocket hit the ground?
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Answer:
• (a)112 feet
,• (b)1.28125 seconds.
,• (c)138.265625 feet.
,• (d)4.22091 seconds
Explanation:
The height of the rocket in terms of the number of seconds t since the rocket's engine stops firing is given below.
[tex]S\mleft(t\mright)=-16t^2+41t+112[/tex]Part A
At the time the rocket stopped firing, t=0.
[tex]S(0)=-16(0)^2+41(0)+112=112[/tex]The rocket was 112 feet above the ground when it stopped firing.
Part B
The value of t at which the rocket reaches its maximum height is the equation of the line of symmetry.
To find this equation, we use the formula below.
[tex]t=-\frac{b}{2a}=-\frac{41}{-2\times16}=1.28125\text{ seconds}[/tex]The rocket reaches its maximum height after 1.28125 seconds.
Part C
To find the maximum height, substitute t=1.28125 into S(t).
[tex]\begin{gathered} S\mleft(t\mright)=-16t^2+41t+112 \\ \implies S(1.28125)=-16(1.28125)^2+41(1.28125)+112 \\ =138.265625\text{ ft} \end{gathered}[/tex]The maximum height of the rocket is 138.265625 feet.
Part D
When the rocket hits the ground, the height is 0.
Set S(t)=0 and solve for t as follows.
[tex]S(t)=-16t^2+41t+112=0[/tex]Using the quadratic formula:
[tex]\begin{gathered} t=\dfrac{-41\pm\sqrt[]{41^2-4(-16)(112)}}{2\times-16}=\dfrac{-41\pm\sqrt[]{1681-(-7168)}}{-32} \\ =\dfrac{-41\pm\sqrt[]{1681+7168}}{-32} \\ =\dfrac{-41\pm\sqrt[]{8849}}{-32} \\ t=\dfrac{-41+\sqrt[]{8849}}{-32}\text{ or }t=\dfrac{-41-\sqrt[]{8849}}{-32} \\ t=-1.658\; \text{or }t=4.22091 \end{gathered}[/tex]Since t cannot be negative, the rocket will hit the ground after 4.22091 seconds.
What is the hcf of 13 and 31?
Answers
Answer: 1
Step-by-step explanation:
The factors of 13 are: 1, 13
The factors of 31 are: 1, 31
So the hcf is 1
shelton earns an hourly wage at a grocery store. the following expression represents Sheltons take home pay after taxes, social security, and his health care plan deducted. let x represent the number of hours shelton worked. 10.25x-0.21(10.25x) part A: which term represents sheltons total pay before deduction. part B:which term represents sheltons deductions. part C:how much is sheltons hourly wage. part D:what percentage is decuted from sheltons pay for taxes , social security , and health care plan. part E: shelton wants to save 1675 for a new laptop. if shelton saves 25% of his take hime pay,how many hours will be need to work to meet his savings goal
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The expression that represents Shelton's take-home pay after all deductions is:
[tex]10.25x-0.21\mleft(10.25x\mright)[/tex](a)Shelton's total pay before deduction = $10.25x
(b) Shelton's deductions = $0.21(10.25x)
(c)Since x represents the number of hours Shelton worked, his hourly wage will be: $10.25
(d)Since 0.21 of his total pay is deducted, the percentage that is deducted from Shelton's pay is 21%.
(e)If Shelton saves 25% of his take-home pay, this will be:
[tex]0.25(10.25x-0.21\mleft(10.25x\mright))[/tex]If he wants to save $1,675, we then have:
[tex]0.25(10.25x-0.21\mleft(10.25x\mright))=1675[/tex]We are required to solve for x.
[tex]\begin{gathered} 0.25(10.25x-2.1525x)=1675 \\ 0.25(10.25x-2.1525x)=1675 \\ 0.25\times8.0975x=1675 \\ 2.024375x=1675 \\ x=\frac{1675}{2.024375} \\ x=827.4\approx827\text{ hours} \end{gathered}[/tex]Shelton will work for 827 hours to meet his goal.
A bouncy ball is dropped such that the height of its first bounce is 6.25 feet and each successive bounce is 74% of the previous bounces height. What would be the height of the sixth bounce the ball
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EXPLANATION
The heigth of the first bounce is 6.25 feet
Each successive bounce is 74% of the previous bounces height
For the first bounce, the ball hit a height of 6.25 feet
The first successive bounce = 74% of the previous bounce
The previous bounce = 6.25 feets
Hence, 74% x 6.25 = Next successive bounce
The next successive bounce = 74/100 x 6.25
The next succesive bounce = 0.74 x 6.25
= 4.625 feets
For the next successive bounce
The previous successive bounce = 4.625 feets
The next successive bounce = 74% x 4.625
The next successive bounce = 0.74 x 4.625
The second successive bounce = 3.4225 feets
For third successive
help with a ab math question
Answers
It seems to be a technical issue with the tool
I can't open the image
Find the indicated probability. Round your answer to 6 decimal places when necessary.Find the probability of tossing 1 tails or 1 head on the first 8 tosses of a "fair" coin.
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First, find the probability of getting 1 head in 8 tosses
P(1 head) = tosses with exactly 1 head/ total number of possible outcomes
number of outcomes = 2^8 = 256
number of outcomes with 1 head = 8 ( we could get 1 head on the first toss or on the second or on the third......)
P (1 head) = 8/256 = 1/32
The same would be true for tails
P(1 tail) = tosses with exactly 1 tail/ total number of possible outcomes
= 8/256 = 1/32
The formula to calculate the “or” probability of two events A and B is this: P(A OR B) = P(A) + P(B) – P(A AND B).
Since we cannot get P(1 head and 1 tail) since we toss 8 times
P (1head or 1 tail) = 1/32 + 1/32 = 2/32 = 1/16 =.0625
7. A large cooler contains the following drinks: 5 lemonades, 9 Sprites, 7 Cokes, and 10 root beers. You randomly pick two cans, one at a time (without replacement). Compute the following probabilities.(a) What is the probability that you get two cans of Sprite? (b) What is the probability that you do not get two cans of Coke? (c) What is the probability that you get either two root beers or two lemonades? (d) What is the probability that you get one can of Coke and one can of Sprite? (e) What is the probability that you get two drinks of the same type?
Answers
A large cooler contains the following drinks,
5 Lemonades, let L reprersent Lemonade
9 Sprites, let S reprersent Sprite
7 Cokes, let C represent Coke and
10 Root beers, let R represent Root beer
Total drinks in the cooler is
[tex]=5+9+7+10=31[/tex]Total outcome = 31 drinks
The formula of probability is
[tex]\text{Probability}=\frac{required\text{ outcome}}{total\text{ outcome}}[/tex]a) The probability that you get two cans of Sprite is
[tex]\begin{gathered} Prob\text{ of picking the first Sprite without replacement is} \\ P(S_1)=\frac{9}{31} \\ Prob\text{ of picking the second Sprite is} \\ P(S_2)=\frac{8}{30} \\ \text{Probability of getting two cans of Sprite}=(PS_1S_2)=\frac{9}{31}\times\frac{8}{30}=\frac{12}{155} \\ (PS_1S_2)=\frac{12}{155} \end{gathered}[/tex]Hence, the the probability that you get two cans of Sprite is 12/155
b)
The probability that you get two cans of Coke
[tex]\begin{gathered} Prob\text{ of picking the first Coke without replacement is} \\ P(C_1)=\frac{7}{31} \\ Prob\text{ of picking the second Coke is} \\ P(C_2)=\frac{6}{30} \\ \text{Probability of getting two cans of Coke is} \\ P(C_1C_2)=\frac{7}{31}\times\frac{6}{30}=\frac{7}{155} \\ P(C_1C_2)=\frac{7}{155} \end{gathered}[/tex]The probability that you do not get two cans of Coke will be
[tex]\begin{gathered} \text{Prob that you do not get two cans of Coke is} \\ 1-P(C_1C_2)=1-\frac{7}{155}=\frac{155-7}{155}=\frac{148}{155} \\ \text{Prob that you do not get two cans of Coke }=\frac{148}{155} \end{gathered}[/tex]Hence, the probability that you do not get two cans of Coke is 148/155
c)
The probability that you get two cans of Root beers is
[tex]\begin{gathered} Prob\text{ of picking the first Root b}eer\text{ without replacement is} \\ P(R_1)=\frac{10}{31} \\ Prob\text{ of picking the second Root b}eer\text{ is} \\ P(R_2)=\frac{9}{30} \\ \text{Probability of getting two cans of Root b}eer\text{ is} \\ P(R_1R_2)=\frac{10}{31}\times\frac{9}{30}=\frac{3}{31} \\ P(R_1R_2)=\frac{3}{31} \end{gathered}[/tex]The probability that you get two cans Lemonades is
[tex]\begin{gathered} Prob\text{ of picking the first Lemonade without replacement is} \\ P(L_1)=\frac{5}{31} \\ Prob\text{ of picking the second Root b}eer\text{ is} \\ P(L_2)=\frac{4}{30} \\ \text{Probability of getting two cans of Lemonade is} \\ P(L_1L_2)=\frac{5}{31}\times\frac{4}{30}=\frac{2}{93} \end{gathered}[/tex]The probability that you get either two root beers or two lemonades is
[tex]P(R_1R_2)+P(L_1L_2)=\frac{3}{31}+\frac{2}{93}=\frac{11}{93}[/tex]Hence, the probability that you get either two root beers or two lemonades is 11/93
d)
[tex]\begin{gathered} Prob\text{ of picking the first Coke without replacement is} \\ P(C)=\frac{7}{31} \\ \text{Prob of picking a can of Sprite is} \\ P(S)=\frac{9}{30} \end{gathered}[/tex]After getting both Sprite and Coke you will multiply the probabilities and then multiply them with 2 because you may choose Coke in first try and Sprite in second or the other way around
The probability that you get one can of Coke and one can of Sprite is
[tex]P(CandS)=2\times(\frac{7}{31}\times\frac{9}{30})=2(\frac{21}{310})=\frac{21}{155}[/tex]Hence, the probability that you get one can of Coke and one can of Sprite is 21/155
e)
Prob of two of each of the cans of drinks (without replacement) are as follow
[tex]\begin{gathered} P(L_1L_2)=\frac{2}{93} \\ (PS_1S_2)=\frac{12}{155} \\ P(C_1C_2)=\frac{7}{155} \\ P(R_1R_2)=\frac{3}{31} \end{gathered}[/tex]The probability that you get two drinks of the same type is
[tex]\begin{gathered} \text{Prob of two drinks of the same type is} \\ =P(L_1L_2)+(PS_1S_2)_{}+P(C_1C_2)+P(R_1R_2) \\ =\frac{2}{93}+\frac{12}{155}+\frac{7}{155}+\frac{3}{31}=\frac{112}{465} \\ \text{Prob of two drinks of the same type}=\frac{112}{465} \end{gathered}[/tex]Hence, the probability that you get two drinks of the same type is 112/465