we have the equation
[tex]\frac{x}{4}=3[/tex]Alondra
Multiply by 4 both sides (property equality of multiplication)
so
[tex]\begin{gathered} \frac{x}{4}\cdot4=3\cdot4 \\ x=12 \end{gathered}[/tex]Courtney
Multiply by 100 both sides (property equality of multiplication)
[tex]\begin{gathered} \frac{x}{4}\cdot100=3\cdot100 \\ 25x=300 \end{gathered}[/tex]therefore
Both followed the teacher's guidelines, but by multiplying by 4, Alondra was able to solve the equation, while Courtney had to apply additional steps.
When purchasing bulk orders of batteries, a toy manufacturer uses this acceptance sampling plan: Randomly select and test 42 batteries and determine whether each is within specifications. The entire shipment is accepted if at most 2 batteries do not meet specifications. A shipment contains 7000 batteries, and 2% of them do not meet specifications. What is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected? The probability that this whole shipment will be accepted is (Round to four decimal places as needed.) The company will accept % of the shipments and will reject % of the shipments, so (Round to two decimal places as needed.)
Solution
Here, we would calculate the probability that one is defective, two are defective, three are defective and the probability that none are defective and sum them together.
The probability that any individual battery is not defective = 1 - 2% = 0.98
We need 42 of them,
Therefore, P(that none are defective) = (0.98)^42 = 0.4281
The probability that a specific battery will be the only defective battery is (0.02)*(0.98)^41
Since we have 42 of them, ((0.02)*(0.98)^41)*42 = 0.367
Note that
[tex]42C2=\frac{42!}{(42-2)!2!}=861[/tex]=> ((0.02)^2*(0.98)^40)*861 = 0.1535
Therefore, 0.4281 + 0.367 + 0.1535 = 0.9486
The probability shows that about 94.86% of all shipments will be accepted
Each of the four graphs below represents a function.Which function has an inverse that is also a function?
For the given question, we will choose the function that has an inverse that will be a function
So, the function must be a one-to-one function
We will use the horizontal line test to check if the function is a one-to-one function or not
the horizontal line will intersect with the graph of the function at only one point
So, the answer will be option 4 as shown in the following figure:
The difference between an integer and its square root is 12. What is the integer?
ANSWER:
16
STEP-BY-STEP EXPLANATION:
From the statement we can establish the following equation (where x is the integer):
[tex]x\:-\:\sqrt{x}\:=\:12[/tex]We solve for x:
This means that the integer that satisfies the statement is 16
what are the roots of the quadratic equation below?[tex]3 {x}^{2} + 9x - 2 = 0[/tex]
Given:
A quadratic equation is:
[tex]3x^2+9x-2=0[/tex]Find-:
The roots of the quadratic equation
Explanation-:
Use quadratic formula:
[tex]ax^2+bx+c=0[/tex]Roots of the equation,
[tex]x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]The roots of the given equation are:
[tex]3x^2+9x-2=0[/tex][tex]\begin{gathered} x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ \\ x_{1,2}=\frac{-9\pm\sqrt{9^2-4(3)(-2)}}{2(3)} \\ \\ x_{1,2}=\frac{-9\pm\sqrt{81-(-24)}}{6} \\ \\ x_{1,2}=\frac{-9\pm\sqrt{81+24}}{6} \\ \\ x_{1,2}=\frac{-9\pm\sqrt{105}}{6} \end{gathered}[/tex]The roots of a quadratic equation are:
[tex]\begin{gathered} x_{1,2}=\frac{-9\pm\sqrt{105}}{6} \\ \\ x_1=\frac{-9+\sqrt{105}}{6},x_2=\frac{-9-\sqrt{105}}{6} \end{gathered}[/tex][tex]\begin{gathered} x_1=\frac{-9+\sqrt{105}}{6},x_2=\frac{-9-\sqrt{105}}{6} \\ \\ x_1=0.2078,x_2=-3.2078 \end{gathered}[/tex]The roots of a quadratic equation are 0.2078 and -3.2078.
For the following figure, complete the statement for the specified pointsRS.199Points RT, S, and areboth collinear and coplanarcolinearcoplanarneither col near nor coplanar
From the figure, we can see that point R, T, S, Q are not on the same plane. Thus we can say they are neither colli coplanar
12 more than the product of 5 and a number x
Answer: (5*x) + 12 or 5x + 12
Step-by-step explanation: differant sites let you do it defferent ways but if the multipaction sign in an "x" I would do the 2nd one :) Hope it's right, have a great day!
which equation shows x^2+6x-4=0 rewritten by completing the squarea) (x+3)^2=36b) (x+3)^2=4c) (x+3)^2=9d) (x+3)^2=13
Solution
Step 1
Write the equation:
[tex]x^2\text{ + 6x - 4 = 0}[/tex]Step 2:
Rewrite the equation:
[tex]x^2\text{ + 6x = 4}[/tex]Step 3
[tex]\begin{gathered} Add\text{ }\frac{b^2}{4a\text{ }}\text{ to both sides to get a perfect square.} \\ \text{a = 1, b = 6} \\ \frac{b^2}{4a}\text{ = }\frac{6^2}{4\times1}\text{ = }\frac{36}{4}\text{ = 9} \end{gathered}[/tex][tex]\begin{gathered} x^2\text{ + 6x + 9 = 4 + 9} \\ Add\text{ similar terms:} \\ (x\text{ + 3\rparen}^2\text{ = 13} \end{gathered}[/tex]Final answer
[tex]d)\text{ \lparen x + 3\rparen}^2\text{ = 13}[/tex]In the figure below, c || d. Classify each of the following angle pairs, and tell whether they arecongruent or supplementary.6. 21 and 23Supp.7. 26 and 23Supp.8. 21 and 283/47/8Supp.5/6t9. 27 and 24Supp.10. 22 and 21& Supp.
Two angles are congruent if they are equal and supplementary if there sum is 180.
Given data:
c and d are parallel.
Now
[tex]\angle1,\angle3[/tex]are corresponding angles, so they are equal.
So,
[tex]\angle1,\angle3[/tex]form a congruent pair.
Now since
[tex]\angle1=\angle6[/tex]since they are vertically oppsoye angles.
And,
[tex]\angle1=\angle3(\text{corresponding angles)}[/tex]So,
[tex]\angle6=\angle3[/tex]So,
[tex]\angle6,\angle3[/tex]form a congruent pair.
Now,
[tex]\begin{gathered} \angle3=\angle8(vertically\text{ opposite angles)} \\ \angle1=\angle3(corresponding\text{ angles)} \\ \Rightarrow\angle1=\angle8 \end{gathered}[/tex]So,
[tex]\angle1,\angle8[/tex]form a congruent pair.
[tex]\angle7=\angle4(vertically\text{ opposite angles)}[/tex]So,
[tex]\angle7,\angle4[/tex]form a congruent pair.
Now,
[tex]\angle1+\angle2=180(linear\text{ pair)}[/tex]So,
[tex]\angle1,\angle2[/tex]form a supplementary pair.
For each triangle, check all that apply. 60° Triangle A 60° Scalene Isosceles O Equilateral 60° 5 Triangle B 11 7 O Scalene O Isosceles O Equilateral 60 triangles by side... Triangle C 30° Scalene Isosceles O Equilateral Triangle D 8 O Scalene O Isosceles O Equilateral
Given:
Four triangles A, B, C and D are given as below
Find:
we have to classify each triangle.
Explanation:
Triangle A:
Since, in the triangle A, all the three angles are equal to 60 degree.Therefore, all three sides of triangle A are equal.
So, Triangle A is Equilateral.
Triangle B:
In triangle B all the three sides are different.
Therefore, triangle B is Scalene.
Triangle C:
In triangle C, all the angles are diffrent, so all three sides are different.
Therefore, Triangle C is Scalene.
Triangle D:
In triangle D, two sides are equal.
Therefore, Triangle D is isosceles
In how many months were there more than two days with thunderstorms? 1 3 5 7
To find how many months have more than 2 days look at the heights
of the bars
You need to count the bars which height more than 2
There are 5 bars that have a height of more than 2
The answer is 5
A standard deck has 52 cards. Half the cards are black and half are red. There are four suits: clubs, spades, hearts, and diamonds. Each suit has 13 cards. There are 4 of each number card, and 4 of each face card (jack, queen, king).Use the image below to help you answer (Part A and Part B)about a deck of cards.PART A: You randomly choose one card from a standard deck. What is the probability that you’ll get a queen? Simplify the fraction.1/161/131/41/2PART B: You randomly choose one card from a standard deck. What is the probability that you will choose a heart? 10%15%25%33%
Pr(queen) = 1/13
Pr(heart) = 25%
Explanation:Part A:
Total cards in a standard deck = 52
from the diagram, there are 4 queens:
1 club queen, 1 spade queen, 1 heart queeen, 1 diamond queen
Probability of getting a queen = number of queens/total number of cards
[tex]\begin{gathered} Pr(queen)\text{ = }\frac{4}{52} \\ Pr(\text{queen) = 1/13} \end{gathered}[/tex]Part B:
from the diagram, there are 13 hearts:
There are 4 suits, each of them has 13 cards. One of the suits is a heart. This means there are 13 cards with hearts
Probability of getting a heart = number of hearts/total number of cards
[tex]\begin{gathered} Pr(\text{heart) = }\frac{13}{52} \\ Pr(\text{heart) = 1/4} \\ In\text{ percentage, 1/4 = }\frac{1}{4\text{ }}\times\text{ 100\% = 25\%} \\ \\ Pr(\text{heart) = 25\%} \end{gathered}[/tex]O DATA ANALYSIS AND STATISTICSFinding the mode and range of a data setEach day, Susan records the number of news articles she reads. Here are her results for the last eight days.3, 6, 6, 9, 1, 6, 9,6Find the range and the mode for the data.
We are given the following data set:
[tex]3,6,6,9,1,6,9,6[/tex]We are asked to determine the range. To do that we will arrange the data from smallest to greatest values, like this:
[tex]1,3,6,6,6,6,9,9[/tex]Now, the range is the difference between the smallest and greatest value, therefore:
[tex]\text{Rage}=9-1=8[/tex]Therefore, the range is 8
Now, the mode is the value that repeats the most. In this case, we have that the value "6" is repeated 4 times, therefore, the mode is:
[tex]\mod e=6[/tex]two cylinders have the same volume the first cylinder has a diameter of 10 cm and a height of 30 cm The second cylinder has a diameter of 8 cm what is the height of the second cylinders the nearest tenth of a centimer
Solution
Given that two cylinders have the same volume but different dimension
For Cylinder 1
Diameter is 10 cm Height is 30 cm
For Cylinder 1
Diameter is 8 cm Height is h cm
The volume of a cylinder is given as;
[tex]V=\pi r^2h[/tex]Since the two cylinders have the same volume,
Since radius = Diameter/2
[tex]\begin{gathered} \pi\times(\frac{10}{2})^2\times30=\pi\times(\frac{8}{2})^2\times h \\ \Rightarrow5^2\times30=4^2\times h \\ \Rightarrow h=\frac{25\times30}{16}\approx46.9 \end{gathered}[/tex]Hence, the second cylinder the nearest tenth of a centimeter is 46.9 cm
Can someone please help me with my math ;( ?
Step 1:
copy and complete the table below by inputting the data provided.
The marginal frequency numbers are the numbers at the edges of the table except for the number at the bottom right-most corner.
B) Marginal relative frequency is the ratio of the sum of the joint relative frequency in a row or column and the total number of data values
Hence,
[tex]\text{ the lowest marginal frequency }=\frac{107}{232}=0.461[/tex]Joint relative frequency the ratio of the frequency in a particular category and the total number of data values
Therefore,
[tex]\text{ the largest joint frequency }=\frac{63}{111}=0.568[/tex]C)For the circled number, the preferred style is Gothic given that the member is male and the percentage is 0.57
D) The total number of males surveyed = 111
the total number of members surveyed = 232
Hence, the percentage of the total number of choir students surveyed that were males are:
[tex]\frac{111}{232}=0.48[/tex]A certain car is rated to get 24 miles per gallon. How far can it travel on 14.5 gallons of gas?
348 miles
Explanation
Step 1
[tex]\text{rate}=24\text{ }\frac{miles\text{ }}{\text{gallon}}[/tex]
it indicates that for every gallon the car can travel 24 miles, Now to know the total miles it can travel with 14.5 gallons of gas, just multiply the rate by the numbers of gallons
so
let
[tex]\begin{gathered} \text{rate}=24\text{ }\frac{miles\text{ }}{\text{gallon}} \\ gallons=14.5 \end{gathered}[/tex]make the product
[tex]\begin{gathered} \text{distace}=\text{rate}\cdot Number\text{ of gallons} \\ \text{replace} \\ \text{distance}=24\frac{miles\text{ }}{\text{gallon}}\cdot14.5\text{ gallons} \\ \text{distance}=348\text{ miles} \end{gathered}[/tex]therefore, the answer is 348 miles
I hope this helps you.
Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.Match each quadratic equation with its solution set.2x^2–8x+5=02x^2-10x-3=02x^2-8x-3=02x^2-9x-1=02x^2-9x+6=0
Solution
For this case we have the following equations:
[tex]2x^2-8x+5=0[/tex]The solutions are:
[tex]x=\frac{4\pm\sqrt[]{6}}{2}[/tex][tex]2x^2-10x-3=0[/tex]Solutions are:
[tex]x=\frac{5\pm\sqrt[]{31}}{2}[/tex][tex]2x^2-8x-3=0[/tex]Solutions are:
[tex]x=\frac{4\pm\sqrt[]{22}}{2}[/tex][tex]2x^2-9x-1=0[/tex]Solutions are:
[tex]x=\frac{9\pm\sqrt[]{89}}{4}[/tex][tex]2x^2-9x+6=0[/tex]Solutions are:
[tex]x=\frac{9\pm\sqrt[]{33}}{4}[/tex]Then final solutions are:
[tex]\frac{9\pm\sqrt[]{33}}{4}\Rightarrow2x^2-9x+6=0[/tex][tex]\frac{4\pm\sqrt[]{6}}{2}\Rightarrow2x^2-8x+5=0[/tex][tex]\frac{9\pm\sqrt[]{89}}{4}\Rightarrow2x^2-9x-1=0[/tex][tex]\frac{4\pm\sqrt[]{22}}{2}\Rightarrow2x^2-8x-3=0[/tex]I need help with homework If angle CVD is 4x-72 and angle BVA is 2x+18, then the value of x is......Also find, angle CVD, angle DVA, angle AVB , angle BVC... I got the picture with the questions
Given,
[tex]\begin{gathered} \angle CVD\text{ = 4x-72} \\ \angle AVB=2x+18 \end{gathered}[/tex][tex]\angle CVD=\angle AVB\text{ (vertically opposite angles.)}[/tex]That is,
[tex]\begin{gathered} 4x-72=2x+18 \\ 2x=90 \\ x=45 \end{gathered}[/tex]Therefore,
[tex]\angle CVD=180-72=108[/tex][tex]\begin{gathered} \angle DVA=180-\angle CVD\text{ (linear pair)} \\ =180-108 \\ =72 \end{gathered}[/tex][tex]\begin{gathered} \angle AVB=2x+18 \\ =90+18 \\ =108 \end{gathered}[/tex][tex]\begin{gathered} \angle BVC=\angle DVA\text{ (vertically opposite angles)} \\ \angle BVC=72 \end{gathered}[/tex]Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. The volleyball team and the wrestling team at Brookfield High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets $2 per car. In addition, they have already brought in $92 from past fundraisers. The wrestling team has raised $16 in the past, and they are making $4 per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?
Data:
Volleyball team: V
Wrestling team: W
x: number of cars
V: $2 per car. Initial $92
W: $4 per car. Initial $16
You have the next equations:
[tex]\begin{gathered} V=2x+92 \\ W=4x+16 \end{gathered}[/tex]To find the total amount you have the next equation:
As each team have raised the same amount:
[tex]\begin{gathered} V=W \\ \\ 2x+92=4x+16 \end{gathered}[/tex]You solve x to find the number of cars each team wash in total:
[tex]\begin{gathered} 2x-4x+92=4x-4x+16 \\ -2x+92=16 \\ \\ -2x+92-92=16-92 \\ -2x=-76 \\ \\ \frac{-2}{-2}x=\frac{-76}{-2} \\ x=38 \end{gathered}[/tex]You use that value of x to find the final amount of each team:
[tex]\begin{gathered} V=2(38)+92=76+92=168 \\ \\ W=4(38)+12=152+12=168 \end{gathered}[/tex]Then, (the total of each team is $168) Total $336, (each team wash 38 cars) The total number of cars 76A student is taking a test in which items of type A are worth 8 points and items of type B are worth 12 points. It takes 3 min to complete each item of type A and 6 min to complete each item of type B. The total time allowed is 60 min and Anna answers exactly 16 questions. How many questions of each type did she complete? Assuming that all her answers were correct, what was her score? She completed questions of type A.
Type A questions are worth 8 points each.
Type B questions are worth 12 points each.
it takes 3 minutes to answer a Type A question
it takes 6 minutes to answer a Type B question.
total time allowed = 60 minutes
She answered a total of 16 questions.
let
number of question answer on type A = x
number of question answered on type B = y
Therefore,
3x + 6y = 60
x + y = 16
then,
[tex]\begin{gathered} 3x+6y=60 \\ x+y=16 \\ x=16-y \\ 3(16-y)+6y=60 \\ 48-3y+6y=60 \\ 3y=60-48 \\ 3y=12 \\ y=\frac{12}{3} \\ y=4 \\ x=16-4 \\ x=12 \end{gathered}[/tex]She answered 12 questions on type A and 4 questions on type B.
If all her answer is correct her score can be computed below
[tex]undefined[/tex]you have two fractions 2/5 and 3/10 and you want to rewrite them so that they have the same denominator what numbers could you use for the denominator is it A 20 or B 10 or C5 or D 15
By definition, a fraction has the following form:
[tex]\frac{a}{b}[/tex]When "a" is the numerator and "b" is the denominator.
In this case you have the following fractions:
[tex]\begin{gathered} \frac{2}{5} \\ \\ \frac{3}{10} \end{gathered}[/tex]Notice that the denominator of the first fraction is 5 and the denominator of the second fraction is 10.
The steps to find the a Common denominator are shown below:
1. Descompose 5 and 10 into their Prime factors:
[tex]\begin{gathered} 5=5 \\ 10=2\cdot5 \end{gathered}[/tex]2. In this case, you need to choose 5, because it is the common one. It will be the Common denominator of the fractions.
3. Divide the original denominator of the first fraction by the Common denominator 5. Multiply the the numerator by the result. Then:
[tex]\frac{2}{5}=\frac{2\cdot1}{5}=\frac{2}{5}[/tex]4. Apply the procedure explained in step 3 to the second fraction:
[tex]\frac{3}{10}=\frac{3\cdot2}{5}=\frac{6}{5}[/tex]5. You can identify that you can also get a common denominator multiplying the denominators and the numerators of both fractions by
The answer is: Option C.
Solve for h by using the inverse (opposite) operations. −5.3+h5=−19.4
Given:
The given term is -5.3+h/5=-19.4.
The objective is to find the value of h using inverse operation.
Inverse operation states that, if a term is in addition operation with another term, then it will change to subtraction operation while shifting to other side of the equation.
If a term is in multiplication operation, then it change to division operation while shifting to other side of the equation.
The value of h can be calculated as,
[tex]\begin{gathered} -5.3+\frac{h}{5}=-19.4 \\ \frac{h}{5}=-19.4+5.3 \\ \frac{h}{5}=-14.1 \\ h=-14.1(5) \\ h=-70.5 \end{gathered}[/tex]Thus, the value of h is -70.5.
Hence, option (c) is the correct answer.
Find the length of the legs.(4,4)[](-2,-2)[?]Enter the number thatbelongs in the green boxEnter
We have a right triangle and we have to find the length of the legs.
We first analyse the hypotenuse.
It is a line that is defined by two points: (4,4) and (-2,-2).
The line passes through the center of coordinates, but we will check it:
The slope can be calculated as:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{4-(-2)}{4-(-2)}=1[/tex]We can now write the slope-point equation as:
[tex]\begin{gathered} y-y_0=m(x-x_0) \\ y-4=1(x-4) \\ y-4=x-4 \\ y=x \end{gathered}[/tex]Then, we know that the equation of the line that defines the hypotenuse is y=x.
If the legs are parallel to the axis, we can find the angle that is defined by the hypotenuse and the horizontal leg as:
[tex]\begin{gathered} \tan (\theta)=m=1 \\ \theta=\arctan (1)=45\degree \end{gathered}[/tex]As we have an angle of 45 degree for one leg, the other has to have an equal angle, as 180-90-45=45 (Note: we are substracting from the sum of the 3 angles, 180 degrees, the already known angles, 90 and 45 degrees, so we are left with 45 degrees that correspond to the third angle).
If the two angles for the legs are equal, their lengths are equal too.
So we can start by calculating the length of the hypotenuse: it is the distance between (4,4) and (-2,-2).
[tex]\begin{gathered} D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt[]{(4-(-2))^2+(4-(-2))^2} \\ D=\sqrt[]{6^2+6^2}=\sqrt[]{2\cdot6^2}=6\sqrt[]{2} \end{gathered}[/tex]Then, we can write the Pythagorean theorem as (a and b are the legs, and c is the hypotenuse):
[tex]\begin{gathered} a^2+b^2=c^2 \\ a=b\longrightarrow2a^2=c^2=(6\sqrt[]{2})^2=36\cdot2=72 \\ a^2=\frac{72}{2} \\ a^2=36 \\ a=\sqrt[]{36} \\ a=b=6 \end{gathered}[/tex]Answer: the length of the legs is 6 units.
what are the first five terms of the recursive sequence aₙ = 3aₙ₋₁ + 3 where a₁ = 9
The expression for the recursive sequence is :
[tex]a_n=3a_{n-1}+3[/tex]where a1 = 9
First term:
Since first term is already given:
[tex]a_1=9[/tex]Second Term :
Substitute n =2 in the recursive expression and simlify
[tex]\begin{gathered} a_n=3a_{n-1}+3 \\ a_2=3(a_{2-1})+3 \\ a_2=3(a_1)+3 \\ a_2=3(9)+3 \\ a_2=27+3 \\ a_2=30 \end{gathered}[/tex]Second Term : 30
Third Term:
Substitute n = 3 in the given recursive expression:
[tex]\begin{gathered} a_n=3a_{n-1}+3 \\ a_3=3(a_{3-1})+3 \\ a_3=3(a_2)+3 \\ a_3=3(30)+3 \\ a_3=90+3 \\ a_3=93 \end{gathered}[/tex]Third Term = 93
Fourth Term:
Substitute n = 4 in the given recursive expression:
[tex]\begin{gathered} a_n=3a_{n-1}+3 \\ a_2=3(a_{2-1})+3 \\ a_2=3(a_1)+3 \\ a_2=3(9)+3 \\ a_2=27+3 \\ a_2=30 \end{gathered}[/tex]
graph f(x)=(x-4)^3-2
Explanation:
Before we plot the graph, you need to understand the transformation process of the parent function x³
x - 4 inside the parenthesis talks about the horizontal shift of the graph. -4 means that the graph will move to the right by 4.
The constant value outside the parenthesis talks about the graph movement. The constant -2 shows that the graph moves downward by 2.
Find the graph of the function below:
hi can someone help me solve this thank you!!
For the given inequality, the smallest value of x is a whole number which is exactly divisible by both 2 and 5.
As given in the question,
Given inequality :
[tex]4x/5 - 1x/2 > 18[/tex]
[tex][2(4x) + 5(-x)]/5.2 > 18[/tex]
[tex](2.4x - 5x) / 5.2 > 18[/tex]
[tex](8x-5x)/10 > 18[/tex]
[tex]3x/10 > 18[/tex]
[tex]3x > 180 \\3x/3 > 180/3\\x > 60\\[/tex]
x ∈ (60, ∞)
∵ Since 60 is whole number and divisible by 2 and 5 both.
therefore answer is option e.
Therefore, for the given inequality, the smallest value of x is a whole number which is exactly divisible by both 2 and 5.
Learn more about inequality here
brainly.com/question/28823603
#SPJ1
Graph triangle ABC with vertices A(0,5) B(4,3) and C(2,-1) and it’s image after a reflection in the line y=2
By reflecting the given points about y=2, the x-coordinates remain the same, then we have
[tex]\begin{gathered} (0,5)\longrightarrow(0,-1) \\ (4,3)\longrightarrow(4,1) \\ (2,-1))\longrightarrow(2,5) \end{gathered}[/tex]then, the new triangle is:
where the red line represents the line y=2. We found the new points by searching the points which are at the same distance of the original poinst to the line:
Then, the preimage and the image after the reflection are:
Craig like to collect vinyl records. Last year he ahead 10 records in his collection. Now he has 12 records. What is the percent increase?
Last year, Craig had 10 records.
Now, he has 12 records.
What is the percent increase?
The percent increase is given by
[tex]\%\: increase=\frac{\text{new value-old value}}{\text{old value}}\times100[/tex]In this case,
Old value = 10 records
New value = 12 records
[tex]\begin{gathered} \%\: increase=\frac{\text{new value-old value}}{\text{old value}}\times100 \\ \%\: increase=\frac{12-10}{10}\times100 \\ \%\: increase=\frac{2}{10}\times100 \\ \%\: increase=20 \end{gathered}[/tex]Therefore, there is a 20% increase in his record collection.
a 2 ft by 2 ft square is divided into smaller squares and portions are shaded. What is the are of the portion and shades portion.?
Answer:
[tex]1.5ft^2[/tex]Explanation:
Here, we want to get the area of the shaded portion
To get this, we need the entire area
The entire area would be the product of the sides of the big square:
[tex]2\text{ }\times2=4ft^2[/tex]Now, let us count the number of shaded small squares.
6 out of 16 squares are shaded
The area of the shaded porion is thus:
[tex]\frac{6}{16}\times4=1.5ft^2[/tex]vertices abc are a(-4,5), b(-2,4), c(-3,2) if abc is reflected across the line y= -2 to produce the image abc; find the coordinates of vertex A
So, the coordinates of the new vertex A must be
[tex](-4,5-14)=(-4,-9)[/tex]So, the coordinates of vertex A is (-4,-9)
Find the derivative of the function y=(4x^ 3 -5x^ 2 )(3x^ 6 +2x^ 5 )10 two different ways. First, multiply the factors togethercollect like terms, then use the basic rules to find the derivative. Second, apply the Product Rule to the function as is currently written.
Given:
The given function is:
[tex]y=(4x^3-5x^2)(3x^6+2x^5)[/tex]First method: Multiplying factors and differentiating after arranging like terms together.
[tex]\begin{gathered} y=(4x^3-5x^2)(3x^6+2x^5) \\ =4x^3(3x^6+2x^5)-5x^2(3x^6+2x^5) \\ =12x^9+8x^8-15x^8-10x^7 \\ =12x^9-7x^{8^{}}-10x^7 \end{gathered}[/tex]Now we will differentiate y with respect to x by basic rules:
[tex]y^{\prime}=12(9)x^{9-1}-7(8)x^{8-1}-10(7)x^{7-1}[/tex]Solving further,
[tex]y^{\prime}=108x^8-56x^7-70x^6[/tex](b) Second method: Apply product rule to find the derivative:
Again,
[tex]y=(4x^3-5x^2)(3x^6+2x^5)[/tex]The product rule states:
[tex](uv)^{\prime}=u^{\prime}v+v^{\prime}u[/tex]Where u and v are the two factors multiplied.
So here we have:
[tex]\begin{gathered} u=4x^3-5x^2 \\ v=3x^6+2x^5 \end{gathered}[/tex]Finding the derivatives:
[tex]\begin{gathered} u^{\prime}=4(3)x^{3-1}-5(2)x^{2-1} \\ =12x^2-10x \end{gathered}[/tex]Similarly,
[tex]\begin{gathered} v^{\prime}=3(6)x^{6-1}+2(5)x^{5-1} \\ =18x^5+10x^4 \end{gathered}[/tex]Now put the values in the product rule,
[tex]\begin{gathered} y^{\prime}=(uv)^{\prime} \\ =u^{\prime}v+v^{\prime}u \\ =(12x^2-10x)(3x^6+2x^5)+(18x^5+10x^4)(4x^3-5x^2) \end{gathered}[/tex]Simplifying further,
[tex]\begin{gathered} y^{\prime}=12x^2(3x^6+2x^5)-10x(3x^6+2x^5)+18x^5(4x^3-5x^2)+10x^4(4x^3-5x^2) \\ =36x^8+24x^7-30x^7-20x^6+72x^8-90x^7+40x^7-50x^6 \\ =108x^8-56x^7-70x^6 \end{gathered}[/tex]This is the derivative obtained.
From above two methods, we can see the derivative is same in both the cases.