Answer:
[tex]\textsf{Factored form}: \quad (a^2+4)(a-2)[/tex]
[tex]\textsf{Standard form}: \quad a^3-2a^2+4a-8[/tex]
Step-by-step explanation:
Given expression:
[tex]\dfrac{a^4-16}{a+2}[/tex]
Rewrite the exponent 4 as 2·2 and 16 as 4²:
[tex]\implies \dfrac{a^{2 \cdot 2}-4^2}{a+2}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{bc}=(a^b)^c[/tex]
[tex]\implies \dfrac{\left(a^2\right)^2-4^2}{a+2}[/tex]
[tex]\textsf{Apply the Difference of Two Squares Formula} \quad x^2-y^2=\left(x+y\right)\left(x-y\right):[/tex]
[tex]\implies \dfrac{(a^2+4)(a^2-4)}{a+2}[/tex]
Rewrite 4 in the second parentheses of the numerator as 2²:
[tex]\implies \dfrac{(a^2+4)(a^2-2^2)}{a+2}[/tex]
Apply the Difference of Two Squares Formula to (a² - 2²):
[tex]\implies \dfrac{(a^2+4)(a+2)(a-2)}{a+2}[/tex]
Cancel the common factor (a + 2):
[tex]\implies (a^2+4)(a-2)[/tex]
Expand:
[tex]\implies a^2(a-2)+4(a-2)[/tex]
[tex]\implies a^3-2a^2+4a-8[/tex]
. a boat costs x dollars, and this cost is to be shared equally by a group of people. in terms of x, how many dollars less will each person contribute if there are 4 people in the group instead of 3
Answer:
x/12
Step-by-step explanation:
In terms of x with no dollar numbers provided, the amount LESS for 4 people versus 3 would be x/3 - x/4. Getting 12 as the LCD and subtracting gives x/12 less to pay.
Amir drove from Jerusalem to the lowest place on Earth, the Dead Sea.
His altitude relative to sea level (in meters) as a function of time (in minutes) is graphed. How long was each descent by 88 meters?
Choose 1 answer:
44
11
10
8
From the given linear function, it is found that each descent by 88 meters was 11 seconds long.
What is the missing information?
The graph of the linear function is missing, and it is given at the end of the answer.
A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.Given two points on the graph, the slope is given by change in y divided by change in x. For this problem, the points are given as follows:
(0, 440) and (40,0).
Hence the slope is:
m = (0 - 440)/(40 - 0) = -11.
Meaning that in each minute, he descended 8 meters. Thus the time to descend 88 meters is given by:
t = 88/8 = 11 seconds.
Each descent by 88 meters was 11 seconds long.
More can be learned about linear functions at https://brainly.com/question/24808124
#SPJ1
Please solve the problem in the image!!!!!
Answer:
455 is the answer of your question
The hotel staff is decorating the lobby by placing one row of string lights along the outer edge of the rectangular ceiling. A member of the staff maps the ceiling on a coordinate grid as shown, where each unit represents 1 meter. What is the approximate length of string lights the staff needs to decorate the ceiling?
The approximate length of string lights the staff needs to decorate the ceiling is 40.25
What is the approximate length of string lights the staff needs to decorate the ceiling?The complete question is added as an attachment
From the attached figure, we have the following coordinates
A = (5, 16)
B = (17, 10)
C = (14, 4)
D = (2, 10)
Calculate the lengths AB and BC using the following distance formula
d = √[(x2 - x1)^2 + (y2 - y1)^2]
This gives
AB = √[(5- 17)^2 + (16- 10)^2] = 13.4
BC = √[(14- 17)^2 + (4- 10)^2] = 6.7
The approximate length of string lights the staff needs to decorate the ceiling is then calculated as
P = 2 * (AB + BC)
This gives
P = 2 * (13.4 + 6.7)
Evaluate
P = 40.2
Hence, the approximate length of string lights the staff needs to decorate the ceiling is 40.25
Read more about perimeter at:
https://brainly.com/question/19819849
#SPJ1
Answer: 40.25
Step-by-step explanation:
Show Your Work
The steps a student took to solve an equation are shown below.
2(x+4)= 6(x-2)
Step 1:
Step 2:
Step 3:
Step 4:
-2x-8=6x-12
-8=4x-12
4 = 4x
x = 1
What error did the student make and what is the correct value
of x?
Answer:
x=5
Step-by-step explanation:
step 1 : exapnding 2(x+4) will give you 2x +8 , the student made an error in expanding
so it should then be 2x+8 = 6x-12
and then if you move the letters to one side and the numbers to the other you will get :
-4x = -20
x=5
hope this works :)
x=5
Step-by-step explanation:
the student added a minus in 2x and 8 when it's suppose to be an addition that is the error
2(x+4)=6(x-2)
2x+8=6x-12
2x-6x=-8-12
-4x=-20
x= -20/-4= 5
x=5
4.4+ 1.5 * 2 / 0.5
Please help!
Answer:
***4.4+ 1.5 * 2 / 0.5*** = 10.4
***3-2.5/3+1 1/3*** = 3.5
Step-by-step explanation:
a hotel chain charges $90 each night for the first two nights and $50 for each additional night's stay. the total cost t is a function of the number of nights x that a guest stays.
The total cost is T(x)={ 90x if 0≤x≤2, 180+50x if x > 2}
What is algebra?Algebra is the branch of mathematics that helps in the representation of problems or situations in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, subtraction, multiplication, and division to form a meaningful mathematical expression.
An algebraic expression in algebra is formed using integer constants, variables, and basic arithmetic operations of addition(+), subtraction(-), multiplication(×), and division(/).
Given:
T(x)={a if x≤2, b if x > 2}
let us take
a= 90x, 0≤x≤2
and, b = 180 + 50x for x>2
Hence. the total cost be T(x)={ 90x if 0≤x≤2, 180+50x if x > 2}
Learn more about this concept here:
https://brainly.com/question/4362480
#SPJ4
Pleae help me with this problem.
Answer:
11) KI = 28
12) PR = 22
Step-by-step explanation:
You want the base segment lengths KI and PR, given relations between those segments and the corresponding midsegment of the triangle.
MidsegmentA midsegment is a line segment that joins midpoints of other line segments, usually the midpoints of triangle sides or opposite sides of a quadrilateral. In the case of a triangle, the midsegment is parallel to the third side, and half its length. This relation is used to solve these problems.
11)KI = 2·AB
(x +37) = 2(x +23) . . . . substitute given expressions
x +37 = 2x +46 . . . . . . eliminate parentheses
37 = x +46 . . . . . . . . . . subtract x
28 = x +37 . . . . . . . . . . subtract 9
The length of KI is 28 units.
12)PR = 2·HG
(x +30) = 2(19 +x) . . . . substitute given expressions
x +30 = 38 +2x . . . . . eliminate parentheses
22 = x +30 . . . . . . . . subtract (x+8)
The length of PR is 22 units.
__
Additional comment
As you can see here, there is no real reason to find the value of x, then substitute it back into the expression for the segment of interest. We can, and did, solve directly for the value of that expression.
As with any algebraic manipulation, any operation performed on one side of the equal sign must also be performed on the other side of the equal sign. When we say "subtract ()", we mean "subtract () from both sides of the equation."
In problem 11, x=-9. In problem 12, x=-8. Negative values of x work just fine, as long as the resulting segment lengths are positive.
find equations of the tangent line and of the normal line to the curve at the point . write the equations of the lines in the form .
The equations of the lines in the form y=mx+b are-
Tangent line: y = 240x - 215994Normal line: y = - 0.00417x + 9.75What is slope?A line's slope is described like the change in y coordinate to the change in x coordinate. The net y coordinate change is y, whereas the net x coordinate change is x.
Therefore the variation in y coordinate in relation to the variation in x coordinate is written as, dy/dx.
Now, as per the given conditions;
The equation of the line is y = (6 + 4x)²
Simplifying the equation;
y = (6 + 4x)²
= 36 + 48x + 16x²
y = 16x² + 48x + 36
Now, differentiate the equation to find the tangent slope;
dy/dx = 32x + 48
At the point (6,900),
dy/dx = 32(6) + 48 = 240
Let equation of the tangent at point (a,b) is;
(y - b) = m(x - a)
a = 6, b = 900, m = 240
y - 6 = 240(x - 900)
Write this in y = mx + b form,
y - 6 = 240x - 216000
y = 240x - 215994 (Equation of Tangent line)
The slope of the normal line = -(1/slope of the tangent line) (since they're both perpendicular to each other)
Slope of the normal line = -1/240
The, equation of the normal will become;
y - 6 = (-1/240)(x - 900)
y - 6 = (-x/240) + 3.75
y = (-1/240)x + 9.75
y = - 0.00417x + 9.75 (Equation of Normal line)
Thus, the equation for the normal and tangent has be found.
To know more about the slope of the line, here
https://brainly.com/question/16949303
#SPJ4
The complete question is-
Find the equation of the tangent line and normal line to the curve y = (6 + 4x)² at the point (6,900). Write the equations of the lines in the form y=mx+b. Tangent line: y=
Normal line: y=
Helpp
Question 3 (Essay Worth 2 points)
(01.02 HC)
Is the expression x3 x3 x3 equivalent to x3 3.3? Why or why not? Explain your reasoning.
No the expression x³.x³.x³ is not equivalent to expression x³ˣ³ˣ³.
Given the expression:
x³.x³.x³
here we apply the law of exponent that, when bases are same powers are added.
Two exponential terms with the same base are multiplied, and while the base doesn't change, their powers are added. On the other hand, their powers are removed when two exponential expressions with the same base are split.
therefore, x³.x³.x³ = x ³⁺³⁺³
= x⁹
and in the second expression x³ˣ³ˣ³.
In this expression powers are multiplied.
= x³ˣ³ˣ³.
= x²⁷
therefore both expressions are not equivalent.
Hence the expressions are not equivalent.
Learn more about Exponents here:
brainly.com/question/847241
#SPJ9
Your set of measuring cups consists of one, a half, a third and a quarter cup measures. If one cup equals 240ml, how many litres does your set hold?
Answer:
0.5L
Step-by-step explanation:
240ml = 1 cup
120ml = 1/2 cup
80ml = 1/3 cup
60ml = 1/4 cup
^ add together and get 500ml, change unit to liters and your set will hold 0.5l
Find the equation of the line that is parallel to y=-2/3x and contains the point (4,-8)
Answer:
y = (-2/3)x + 16/3
Step-by-step explanation:
Any line parallel to y = (-2/3)x has the same slope: (-2/3).
Then, in this pattern, y = (-2/3)x + c, where c is the y-intercept.
If (4, -8) lives on this new line, then:
-8 = (-2/3)(4) + c, and
-8 = -8/3 + c
This last equation is equivalent to -24/3 + 8/3 + c => -16/3 = -c.
Therefore the y-intercept is 16/3 = c, and the desired equation is
y = (-2/3)x + 16/3
Write an absolute value equation that has the solutions x=-6 and x=10.
Answer:
[tex]r \times \cos( 0) = - 6 \\ r = - \frac{6}{ \cos(0) } \\ r = - 6 \sec(0) \\ \\ r \times \cos(0) = 10 \\ r = \frac{10}{ \cos(0) } \\ r = 10 \sec(0) [/tex]
Two different numbers are selected ffrom -2,-2,0,3,4,5 and multiplied together. what is the probability that the product is zero
Two numbers are drawn without replacement. The only way their product is zero is if one of these numbers is zero. The probability of this happening is
[tex]\dfrac{\dbinom11 \dbinom51}{\dbinom62} = \dfrac{1\cdot5}{15} = \boxed{\dfrac13}[/tex]
In other words,
• there is 1 zero in the list - we draw it with [tex]\binom11 = 1[/tex] way of doing so;
• there are 5 non-zero numbers in the list - we draw 1 of these, with [tex]\binom51 = 5[/tex] ways of doing so;
• there is a total of 6 numbers in the list - we draw 2 of these, with [tex]\binom62 = 15[/tex] ways of doing so.
On the other hand, if the numbers are drawn with replacement and independently of one another, then each number has the same probability of getting drawn each time. There are 6×6 = 36 possible outcomes, each with 1/36 probability of occurring. Their product is zero if at least one of the numbers drawn is zero. The probability of this happening is
[tex]\dbinom21 \dfrac16 \times \dfrac56 + \dbinom22 \left(\dfrac16\right)^2 = \dfrac{11}{36}[/tex]
That is,
• there is 1/6 chance of drawing a zero and 5/6 chance of not drawing a zero. This can selection can happen in [tex]\binom21=2[/tex] ways (zero is drawn first or last);
• there is 1/6 chance of drawing a zero for either selection. This can happen in [tex]\dbinom22=1[/tex] way (zero is drawn both times).
While the probabilities are close, they are not the same! Be careful with which interpretation you choose.
[tex]\dfrac13 = \dfrac{12}{36} > \dfrac{11}{36}[/tex]
In a classroom, 14 children have a dog as a pet, 11 children have cats and 3 who have both. If 24 children don't have either a cat of a dog, how many children are in the classroom.
Greetings from Brasil...
Using set theory to solve this question
Let
D = Dogs Set
C = Cats Set
R = Classroom Set
n(D ∪ C) = n(D) + n(C) - n(D ∩ C)
n(D ∪ C) = 14 + 11 - 3
n(D ∪ C) = 22
Then,
Total = n(D ∪ C) + 24
Total = 22 + 24
Total = 46Question 3
Examine the table, which contains some points of a quadratic function.
X
-1
0
1
2
3
Which
f(x)
12
15
16
15
12
mont correctly describes the end behavior of the function?
Answer
this question is impossible!11!1!!!!
Step-by-step explanation:
none
(PLEASE I NEED THIS) Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set.
y<2x-8
y>-1/2-3
The solution of the graph of the solution set y< 2x - 8 and y > -1/2 x - 3 is shown in image by purple shade.
What is Inequality?
To compare two or more mathematical expression in a relation is called an Inequality.
Given that;
The solution sets are;
y < 2x - 8
y > -1/2 x - 3
Now, We draw graph of inequalities.
Then we get,
The graph of the solution set y< 2x - 8 and y > -1/2 x - 3 is shown in image by purple shade.
Learn more about the inequality visit:
https://brainly.com/question/11234618
#SPJ1
Use the position equation given below, where s represents the height of the object (in feet),
v0
represents the initial velocity of the object (in feet per second),
s0
represents the initial height of the object (in feet), and t represents the time (in seconds), as the model for the problem.
s = −16t2 + v0t + s0
You drop a coin from the top of a building. The building has a height of 1028 feet.
(a) Use the position equation to write a mathematical model for the height of the coin.
s = ?
(b) Find the height of the coin after 1.5 seconds.
s = ? ft
(c) How long does it take the coin to strike the ground? (Round your answer to two decimal places.)
t = ? sec
(I really can't comprehend how to due this but its due soon, please help me!)
Using the motion equation we can get:
a) s = -16*t^2 + 1028
b) 992 feet above the ground.
c) after 8 seconds, the coin will hit the ground.
How to write the height equation?
First, we know that the general height equation is:
s = -16*t^2 + v0*t + s0
In this case, the coin is dropped, so the initial velocity v0 is zero.
And the coin is dropped from a height of 1028 ft, then s0 = 1028
Replacing that in the equation we get:
s = -16*t^2 + 1028
b) The height after 1.5 seconds is what we get by evaluating the height equation in t = 1.5
s = -16*(1.5)^2 + 1028 = 992
This means that the height after 1.5 seconds is 992 ft above the ground.
c) The coin will strike the ground when s = 0, then we need to solve:
s = 0 = -16*(t)^2 + 1028
Solving that for t, w eget:
16*t^2 = 1028
t^2 = 1028/16
t = √(1028/16) = 8
This means that the coin will hit the ground after 8 seconds.
If you want to learn more about motion equations:
https://brainly.com/question/19365526
#SPJ1
y−t/x=m Solve for 'y'.
Answer:
y = m(x) + t
Step-by-step explanation:
Pretty sure this is the right answer...
Hope this helps! :D
a team can row 40km in 2 hours when rowing with the current, but only 16km in 2 hours when rowing against the current. determine the teams rowing speed when there is no current
Based on the rowing speed of the team when rowing with the current and when rowing against, the team rowing speed when there is no current is 14 km/h.
What is the rowing speed without current?Assuming that x is the rowing rate and y is the rate of the current, the equations needed will be:
2 (x + y) = 40
2 (x - y) = 16
We can add the x-values of the equation to find the value of x which will be the rowing speed without current:
2x + 2x = 40 + 16
4x = 56
x = 56 / 4
x = 14 km /h
Find out more on rowing speed at https://brainly.com/question/13492125
#SPJ1
the length of a rectangle is four times its width. if the perimeter is at most 106 centimeters, what is the greatest possible value for the width? which inequality models this problem?
If you solve the inequality you will have a final answer w ≤ 10. The greatest value being 10.
According to the statement
We have to find that the value of the width.
So, For this purpose, we all know that the
A rectangle may be a form of quadrilateral, whose opposite sides are equal and parallel.
From the given information:
the length of a rectangle is fourfold its width. if the perimeter is at the most 106 centimeters
Then
L = 4w --------The length of a rectangle is fourfold its width.
2w + 2L ≤ 106 ------- the perimeter is at the most 130 centimeters.
Now, if substitute the primary equation into the second inequality you may get
2w + 2 • (4w) ≤ 106.
Therefore, the inequality model becomes 2w + 2 • (4w) ≤ 106.
So, If you solve the inequality you will have a final answer w ≤ 10. The greatest value being 10.
Learn more about rectangle here
https://brainly.com/question/19819849
#SPJ4
5. Owen walked 5 miles in 4 hours. Fill out a table of
equivalent ratios and plot the points on the coordinate axes
provided.
F
Miles
5
40
Hours
4
36
6
n
P
Table:
Miles Hours
5 4
40 32
45 36
The ratio between Miles and Hours is 5:4
In the table, miles = 40 is got by 8 x 5 = 40
So corresponding hours = 8 x 4 = 32
In the table, hours = 36 is got by 9 x 4 = 36
So corresponding miles = 9 x 5 = 45
So the ratios are 5:4 , 40:32, 45:36
Graph can be plotted with number of miles in the x-axis and number of hours in y-axis.
The points to be plotted are (5, 4), (40,32) and (45, 36). Joint these points to get a straight line on the graph.
Learn more about ratios at https://brainly.com/question/2328454
#SPJ1
8y-2(y+4) pls help wil be rewarded with prize
Answer:
[tex] \large \sf \: 6y - 8[/tex]
Step-by-step explanation:
Let's simplify step-by-step.
8y−2(y+4)
Distribute:
=8y+(−2)(y)+(−2)(4)
=8y+−2y+−8
Combine Like Terms:
=8y+−2y+−8
=(8y+−2y)+(−8)
=6y+−8
=6y−8
Answer:
6y - 8
Step-by-step explanation:
This is a simplifying problem, so we just need to separate and link the like terms.
First we start with the parenthesis by distributing
-2*y + -2*4
-2y - 8
We can the put it back into the equation
8y - 2y + 8
8y and -2y are like terms so we can put them together to get 6y
6y +8
the set of all real numbers less than 50 or less than 50 set builder notation
Here is a simple example of set-builder notation set builder notation it says " the set of all x's, such that X is greater than 0".
select the points you need to calculate the average rate of change from the beginning of the slide to when the slide has covered a horizontal distance of 15 feet. (0, 80) (5, 40) (10, 20) (15, 10) the average rate of change is .
The typical pace of change since the slide's start, -14/3.
Let, the average rate of change is k
(x1,y1) (x2,y2), k= (y2-y1)/(x2-x1)
=10-80/15
=-70/15
k=-14/3
Average = -14/3
How do you find average rate?Divide the change in y-values by the change in x-values to determine the average rate of change. Identifying changes in measurable parameters like average speed or average velocity calls for the knowledge of the average rate of change.
The average rate of change is a measure of how quickly on average one quantity changes in relation to another. It provides a general sense of the function's change in value per unit throughout the specified time period. Simply said, the rate of change is calculated by dividing the amount of change in one item by the equal amount of change in another. Let's take a closer look at the average rate of change formula.
To know more about average rate visit:
https://brainly.com/question/96116
#SPJ4
Solve the equation by graphing. If the exact roots cannot be found, state the consecutive integers between which the roots are located. If there are no roots, state no real solution. List all roots or consecutive integers in order from least to greatest. 2 x^ 2 + x = 11
The roots of the quadratic equation represented by the graph are; x = -2.5 and x = 2.5.
What is the Solution of the Quadratic Equation?
The solutions to the equation 2x² + x = 11 will be given by the points at which the graph intercepts the x-axis.
By looking at the graph, we can clearly see that the graph intercepts the x-axis at x = -2.5 and x = 2.5.
The roots are located between -3 and 3.
Read more about Solution of Quadratic equations as; https://brainly.com/question/11347015https://brainly.com/question/12952549
#SPJ1
Ross is buying pecans and cashews at the supermarket. There is a proportional relationship between the weight of the nuts and the cost.
This table shows the cost of pecans:
Pecans
Weight (pounds) Cost
0.5 $5
1.5 $15
4 $40
This equation describes the cost of cashews, where y represents the cost in dollars and x represents the weight in pounds:
y=9.35x
Which nuts have a higher unit price?
How much more do those nuts cost?
$
more per pound
The nut that has the higher unit price is pecans.
The pecans cost $0.65 more than cashews.
Which nut cost more?Looking at the table of the cost of pecan, it can be seen that the cost of pecan increase by $10 for every 1 pound increase in weight.
Cost of one pound of pecan : $5 / 0.5 = $10
A linear function is a function that has one dependent variable and one independent variable. A linear function represents a straight line on a coordinate plane.
A linear function has the form: a + bx
Where:
a = constant b = independent variabley = 9.35x
Cost of one pound of cashew : 9.35 x 1 = $9.35
Difference in cost = $10 - $9.35 = $0.65
To learn more about linear functions, please check: https://brainly.com/question/26434260
#SPJ1
an online retailer is shipping a shower curtain pole that is 13 feet long. their largest box is in the shape of a rectangular prism that is 10 ft × 9 ft × 8 ft. will the pole fit in the box?
Yes, the pole will fit in the box when kept diagonally.
In order to fit the pole in rectangular prism or cuboid, it has to be kept diagonally. Now, we need to calculate the length of body diagonal of cuboid. The length of body diagonal of cuboid should be greater than pole to fit it.
The length of body diagonal of cuboid can be calculated by the formula -
Diagonal = √(l² + b² + h²)
Keep the values in formula to find the diagonal of rectangular prism
Diagonal = √(10² + 9² + 8²)
Taking square of the three numbers to find the diagonal of rectangular prism
Diagonal = √100 + 81 + 64
Taking sum of numbers present on Right Hand Side of the equation
Diagonal = √245
Taking square root of the number
Diagonal = 15.65 feet
Hence, as the diagonal of rectangular prism is greater than shower curtain pole, it will definitely fit.
Learn more about diagonal calculation -
https://brainly.com/question/18839786
#SPJ4
Given the figure below, find the values of x and z.
20
X
(5x + 39)°
71°
Answer:
Step-by-step explanation:
(5x+39)° and 71° are in a straight line therefore their summation will have to give us 180°(supplementary angles)5x+39°+71°=180°5x=180°-39°-71°5x=70°[tex]\frac{5x}{5} =\frac{70}{5} \\x=14\\[/tex] z is vertically opposite to 71° therefore z= 71°Convert the following decimal into a fraction.
0.83
Answer:
83/100
Step-by-step explanation:
To turn a decimal into a fraction, place decimal number over its place value.
Place value = 100
Decimal number = 83