3(x + 10) < 2 (20 – x)
Distributing multiplication over the addition and the subtraction, we get:
3x + 3*10 < 2*20 - 2x
3x + 30 < 40 - 2x
30 is adding on the left, then it will subtract on the right.
2x is subtracting on the right, then it will add on the left.
3x + 2x < 40 - 30
5x < 10
5 is multiplying x on the left, then it will divide on the right.
x < 10/5
x < 2
(a) Name the vertex (b) Which direction does the graph open? (c) Find the Axis of Symmetry (d) Graph (e) Maximum or Minimum (f) Name the domain and range. (9) Write the equation in standard form
The vertex of the parabolic equation y = (x+2) (x-2) is at (0,4) .
The given equation is of the form y = (x+2) (x-2).
This can be simplified using the properties of algebra:
y = x² - 4
the graph of the parabolic function y = x² - 4 is attached below.
a) The vertex of this graph is at (0,4) .
b) The graph opens upwards
c)The axis of symmetry of the graph is the y-axis or the line x = 0
e)For finding the maximum and minimum values we find the derivative of x.
y = x² - 4
dy/dx = 2x
at dy/dx = 0 , x = 0
at x=0 , y= - 4
hence the function has a minimum value at -4 and the maximum value of y is infinity.
f) Domain of the function (-∝ < x < ∝)
range of the function ( y ≥ 4 )
g) The equation in the standard form is given by y = x² - 4 .
Disclaimer: The missing equation is y = (x+2)(x-2).
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write a function to describe the following scenario.a garden watering bucket has 3,000 mm of water in it but there is a hole that is leaking 18 mm every minute how much water remains in the container after certain number of minutes? y= ? - __x
The bucket has 3000 milliliters of water.
It leaks 18ml per minute.
Let "y" represent the remaining water after a certain number of minutes and "x" represent the number of minutes passed.
"3000ml" represents the y-intercept of the function (the amount of water in the bucket at x=0 minutes)
and "-18ml" represents the amount of water that has leaked after a certain amount of time, and is the slope of the function. The value is negative because the volume of the bucket is decreasing as time passes.
Then the function will be
[tex]y=3000-18x[/tex]WILL GIVE BRAINLEST! I NEED HELP ASAPPP! The highest score on an Algebra test was 40 points more than the lowest. When added together, the lowest and highest score was 152. Write an equation to find the highest score, then solve.
A= x + x + 40 = 152; 56
B= x + x = 152; 76
C= x + x - 40 = 152; 96
D= x + x + 40 = 152; 96
Answer:
D is correct.
Step-by-step explanation:
Let x be the lowest score. Then x + 40 is the highest score.
[tex]x + x + 40 = 152[/tex]
[tex]2x + 40 = 152[/tex]
[tex]2x = 112[/tex]
[tex]x = 56[/tex]
[tex]x + 40 = 96[/tex]
Lowest score is 56, highest score is 96.
As part of an art installation, Larry wants to tie a piece of rope from the end of a 10m branch that is sticking out of the ground at 23.58 degrees angle to the end of a 1 m tall stake that is 4 m west of the end of the branch. The plans for the installation are shown below. Answer the questions below:1. How many meters above the ground is the end of the branch?2. What is the shortest possible rope length (in meters) that Larry can use to attach the end of the branch to the top of the stake?
Given data:
The length of branch is BC=10 m.
The given angle is β=23.58 degrees.
1)
The expression for the height of the branch above the ground is,
[tex]h=BC(\sin \beta)[/tex]Substitute the given values in the above expression.
[tex]\begin{gathered} h=(10m)(sin(23.58^{\circ})) \\ =4m\text{ } \end{gathered}[/tex]Thus, the height of the branch above the ground is 4 m.
10. The perimeter of the rectangle to the right is 28 ft. What is the value of x?
ANSWER
x = 9
EXPLANATION
We have the rectangle with width 3 ft and length (2 + x) ft.
The perimeter of the triangle is 28 ft.
The perimeter of a rectangle is given as:
P = 2(L + W)
where L = length
W = width
Therefore, we have that:
28 = 2[(2 + x) + 3]
28 = 2(2 + x + 3) = 2(x + 5)
28 = 2x + 10
=> 2x = 28 - 10 = 18
Divide through by 2:
2x/2 = 18/2
x = 9
That is the value of x.
help me please!! (10 pts)
(2,3) (4,6) (6,9) (8,12) is the set of ordered pair lie on the function that is direct proportion.
Direct proportion is mathematical comparison between two variable
when one increase also increase the other or one decrease also decreases the other then , they are direct proportion.
In direct proportion , the ratio of these variable remains same no matter what.
The following are the set of ordered pair,
a. (2,6) (4,8) (6,10) (8,12)
calculating ratio,
[tex]\frac{2}{6} = \frac{1}{3} \neq \frac{4}{8} = \frac{1}{2}[/tex]
Ratio is changing so, ordered pair are not direct proportion
b. (2,2) (4,2) (6,2) (8,2)
ratios = [tex]\frac{2}{2} =1 \neq \frac {4}{2} = 2[/tex]
Ratio is different
c. (2,1) (4,3) (6,5) (8,7)
Ratio is different , the ordered set is not direct proportion
d. (2,3) (4,6) (6,9) (8,12)
ratios = [tex]\frac{2}{3}=\frac{4}{6}[/tex]
Ratios are same in entire ordered set
Hence , (2,3) (4,6) (6,9) (8,12) is a direct proportion.
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The graph of a quadratic function with vertex (-1,4) is shown in the figure below. Write the domain and range in interval notation.
Background:
• Domain,: a set of all possible values of the independent variable (,x,, in this case).
,• Range,: a set of all possible values of the dependent variable (,y,, in this case), after substituting the domain.
Based on the arrows of the function, we can conclude that those extend to infinity (negative and positive).
Also, based on the coordinates of the vertex given we can see that the first value of y is 4.
Answer:
• Domain
[tex](-\infty,\infty)[/tex]• Range
[tex](4,\infty)[/tex]The sum of two numbers is 200 and their difference is 28.What are the two numbers?
Let us assume the numbers are x and y.
The first part of the question can be written as
[tex]x+y=200\text{ ---------------(1)}[/tex]and the second part can be written as
[tex]x-y=28\text{ --------------(2)}[/tex]From equation 1, we can get a value for y as
[tex]y=200-x\text{ -------------(3)}[/tex]Substitute for y in equation 3 into equation 2:
[tex]x-(200-x)=28[/tex]Expanding and solving, we get
[tex]\begin{gathered} x-200+x=28 \\ 2x=200+28 \\ 2x=228 \\ \therefore \\ x=\frac{228}{2} \\ x=114 \end{gathered}[/tex]Next, we substitute for the value of x into equation 3:
[tex]\begin{gathered} y=200-114 \\ y=86 \end{gathered}[/tex]Therefore, the two numbers are 114 and 86
Using the slope formula, find the slope of the line through the points (0, 0) and (5, 20).
The slope formula for 2 points is
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]where
[tex]\begin{gathered} (x_1,y_1)=(0,0) \\ (x_2,y_2)=(5,20) \end{gathered}[/tex]By substituting these values into the slope formula, we get
[tex]\begin{gathered} m=\frac{20-0}{5-0} \\ m=\frac{20}{5} \\ m=4 \end{gathered}[/tex]therefore, the slope is 4.
At t seconds after launch is given by the function… how long will it take the rocket to reach its maximum height? What is the maximum height?
Given:
The height equation is,
[tex]h(t)=-16t^2+144t+6[/tex]Explanation:
For maximum/minimum of a function, the first derivative of function is 0.
Differentiate the function with respect to x.
[tex]\begin{gathered} \frac{d}{dt}h(t)=\frac{d}{dt}(-16t^2+144t+6) \\ =-32t+144 \end{gathered}[/tex]For maximum and minimum,
[tex]\begin{gathered} -32t+144=0 \\ t=\frac{144}{32} \\ =4.5 \end{gathered}[/tex]So rocket reach it maximum height after 4.5 seconds of launch.
Substitute 4.5 for t in the equation to determine the maximum reached by rocket.
[tex]\begin{gathered} h(4.5)=-16(4.5)^2+144\cdot4.5+6 \\ =-324+648+6 \\ =330 \end{gathered}[/tex]So maximum height of rocket is 330 feet.
find the upper and lower sums for the region bounded by the graph of the function and the x-axis on the given interval. Leave your answer in terms of n, the number of subintervals.
Explanation
The area under a curve between two points can be found by doing a definite integral between the two points
Step 1
a) set the intergral
[tex]\begin{gathered} limits:\text{ 1 and 2} \\ function:\text{ f\lparen x\rparen=6-2x} \end{gathered}[/tex]hence
[tex]Area=\int_1^26-2x[/tex]Step 2
evaluate
let ; numbers of intervals
[tex]\begin{gathered} \begin{equation*} \int_1^26-2x \end{equation*} \\ \int_1^26-2x=\lbrack6x-x^2\rbrack=(12-4)-(6-1)=8-5=3 \end{gathered}[/tex]therefore, the area is
[tex]area=3\text{ units }^2[/tex]
I hope this helps you
14. John rides his motorcycle for 0.2 hours with a constant speed of 68 km/h and then foranother 13 minutes with a constant speed of 102 km/h. What is his average speed for thetotal trip?
We must calculate the weighted average as follows:
[tex]\begin{gathered} \frac{68\cdot0.2+102\cdot\frac{13}{60}}{0.2+\frac{13}{60}} \\ \frac{13.6+22.1}{0.416}=85.68 \\ \end{gathered}[/tex]Therefore, the average speed is 85.68 km/h
In O O, mCD = 30° and CA BD. Also, the center of the circle,point o, is the intersection of CB and AD.DWhat is mAB?АBMAB = Just answers
Answer:
mAB = 30°
Explanation:
mCD is the measure of angle 1 and mAB is the measure of angle 2. These angles are vertically opposite because they are formed by intersecting lines and they are on opposite sides.
Vertically opposite angles have the same measure, so:
mAB = mCD
mAB = 30°
Therefore, the measure of AB is 30°
Find a quadratic function of the form y=ax^2 that passes through the point (-2,-8)
Solution
[tex]\begin{gathered} \text{ since }y=ax^2 \\ \\ \text{ at }(-2,-8) \\ \\ \Rightarrow-8=a(-2)^2 \\ \\ \Rightarrow-8=a(4) \\ \\ \Rightarrow a=-\frac{8}{4}=-2 \\ \\ \Rightarrow y=-2x^2 \end{gathered}[/tex]The quadratic equation is
[tex]y=-2x^2[/tex]the circumference of a circular garden is 109.9 feet. what is diameter of the garden? use 3.14 and do not round your answer.
The circumference of a circle is given by the formula:
[tex]C=\pi d[/tex]Where d is the diameter of the circle.
If the circumference is 109.9 ft, we have:
[tex]\begin{gathered} 109.9=3.14\cdot d \\ d=\frac{109.9}{3.14} \\ d=35\text{ ft} \end{gathered}[/tex]So the diameter of the garden is 35 feet.
what the lowest terms for 15/75
the given expression is
15/ 75
that is
1/5
thus, the lowest term is 1/5
I need help with this problem if anyone want to help me please do thanks
Solve e from the equation by substraction 96 to both sides of the equal sign:
[tex]undefined[/tex]Triangle RJM has an area of 6 and a perimeter of12. If the triangle is dilated by a scale factor of 3centered at the origin, what are the area andperimeter of its image, triangle R'I'M"?1) area of 9 and perimeter of 152) area of 18 and perimeter of 363) area of 54 and perimeter of 364) area of 54 and perimeter of 108
The perimeter of a triangle is given by:
[tex]P=s_1+s_2+s_3=12[/tex]Now, if the triangle is dilated by a factor of 3 this means that we multiply each side by 3, then we have:
[tex]P^{\prime}=3s_1+3s_2+3s_3=3(s_1+s_2+s_3)=3(12)=36_{}_{}[/tex]Therefore the new triangle will have a perimeter of 36.
Now, the original area is given by:
[tex]A=\frac{1}{2}bh=6[/tex]if we dilate the triangle by a factor of three we get:
[tex]A=\frac{1}{2}(3b)(3h)=9(\frac{1}{2}bh)=9(6)=54[/tex]Therefore the new area is 54.
Wiith this we conclude that the answer is 3.
Using the figure below as a starting point, identify the figure in which lines to l are drawn through points A, B, C, and D.
SOLUTION
We want to find the figure in which lines perpendicular to l are drawn through points A, B, C, and D
The correct figure will be the one in which a vertical line is drawn across each of points A, B, C and D.
Looking at this, we can see that the correct answer is the first option
Answer:
a
Step-by-step explanation:
Construct triangle ABC if AB = 5cm, BC=5cm and AC=3cm. What type if triangle does this create
The type of triangle is isosceles.
Picture
A certain loan program offers an interest rate of 4%, compounded continuously. Assuming no payments are made, how much would be owed after six yearson a loan of I300Do not round any intermediate computations, and round your answer to the nearest cent
In order to calculate how much will be owed, we can use the formula below for interest compounded continuously:
[tex]A=P\cdot e^{rt}[/tex]Where A is the final amount after t years, P is the initial amount and r is the interest rate.
So, using P = 1300, r = 0.04 and t = 6, we have:
[tex]\begin{gathered} A=1300\cdot e^{0.04\cdot6}\\ \\ A=1300\cdot e^{0.24}\\ \\ A=1652.62 \end{gathered}[/tex]Therefore the amount owed after 6 years is $1652.62.
What specific measure of a geometric figure is shown in the image?90100 110 12080 70 6013014049 15030 40 50 60 70 80140 130 120 110 10070 20 1010 170160 170 1800 i10294 5 632-N5LLLLLLLLگی۔LEO A. A 140 mm side lengthO B. A 40° angleO C. A 140° angleOO D. A 180 mm side length
ANSWER :
The answer is C. a 140-degree angle
EXPLANATION :
A protractor is used to measure the angle.
From the figure, the angle is 140 degrees.
a right triangle is shownwhich angle measure is closet to x
We know that
[tex]\cos (x)=\frac{20}{24}[/tex]Solving for x,
[tex]\begin{gathered} x=\cos ^{-1}(\frac{20}{24}) \\ \Rightarrow x=33.56 \end{gathered}[/tex]x is aproximately 33.56
In the diagram below AB⊥CD and bisects ∠MOP.(a) If m∠MOP=130° find m∠POD.(b) If m∠COM=38°, find m∠MOP and m∠POD.
A
Since AB in perpendicular to CD and bisects mThis can be written as
[tex]m<\text{MOP}+m<\text{COM}+m<\text{DOP}=180\text{ (1)}[/tex]but
[tex]m<\text{COM}=m<\text{DOP}[/tex]then
[tex]m<\text{MOP}+2m<\text{DOP}=180[/tex]pluggin the value of the angle m[tex]\begin{gathered} 130+2m<\text{DOP}=180 \\ 2m<\text{DOP}=180-130 \\ m<\text{DOP}=\frac{50}{2} \\ m<\text{DOP}=25 \end{gathered}[/tex]Therefore the angle m
B
As we mentioned above the angle mthen m
Using equation (1) of part to find the angle m[tex]\begin{gathered} m<\text{MOP}+38+38=180 \\ m<\text{MOP}=180-76 \\ m<\text{MOP}=104 \end{gathered}[/tex]therefore the angle m
Solve the radical equation.9-6=27-29What is the extraneous solution to the radical equation?O 1O 9Both 1 and 9 are extraneous solutions to the equation.O There are no extraneous solutions to the equation.
Given the radical equation:
[tex]q-6=\sqrt[]{27-2q}[/tex]Squaring both sides to eliminate the root.
[tex]\begin{gathered} (q-6)^2=27-2q \\ q^2-12q+36=27-2q \\ q^2-12q+2q+36-27=0 \\ q^2-10q+9=0 \end{gathered}[/tex]Factor the equation to find the roots:
[tex]\begin{gathered} (q-1)(q-9)=0 \\ q-1=0\rightarrow q=1 \\ q-9=0\rightarrow q=9 \end{gathered}[/tex]we will check ( q = 1 and q = 9 ) by substitution into the given equation:
When q = 1
[tex]\begin{gathered} q-6=1-6=-5 \\ \sqrt[]{27-2q}=\sqrt[]{27-2}=\sqrt[]{25}=5 \end{gathered}[/tex]So, ( q = 1 ) is an extraneous solution.
When q = 9
[tex]\begin{gathered} q-6=9-6=3 \\ \sqrt[]{27-2q}=\sqrt[]{27-18}=\sqrt[]{9}=3 \end{gathered}[/tex]So, ( q = 9 ) is the solution of the given equation.
So, the answer will be:
The extraneous solution to the radical equation is 1
Given Point A, what is the coordinate for A' after the following transformation has occurred?LaTeX: \left(x,y\right)\rightarrow\left(x-5,\:-y+2\right)A (5, 7)Al.
Given:
The point A(5, 7).
To given transformation is (x-5, -y+2).
So,
The new point is,
[tex]A^{\prime}(5-5,-7+2)=A^{\prime}(0,-5)[/tex]Therefore, the coordinate for A' after the given transformation has occured is A'(0,-5).
Relationship A has a greater rate than Relationship B. This table represents Relationship B.Hours worked2458Amount paid30.4060.8076121.60Which equation could represent Relationship A?Hours worked is represented by x and Amount paid is represented by y.Select each correct answer.y = 15.4xy = 15.2xy = 16.4xy = 14.9x
ANSWER:
[tex]\begin{gathered} y=15.4x \\ y=16.4x \end{gathered}[/tex]STEP-BY-STEP EXPLANATION:
We can calculate the equation that represents relationship B, calculating the slope using the data from the table, like this:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ \text{replacing} \\ m=\frac{121.60-30.40}{8-2} \\ m=\frac{91.2}{6} \\ m=15.2 \end{gathered}[/tex]Therefore, the equation of relationship B is:
[tex]y=15.2x[/tex]Therefore, the relationship A, having a greater rate, could be the following:
[tex]\begin{gathered} y=15.4x \\ y=16.4x \end{gathered}[/tex]Assume the given function is one-to-one. Find the indicated value:If f(3)=2 then f^{-1}(2)=AnswerIf f^{-1}(-2)=-1 then f(-1)=?Answer
The question asks us to perform the inverse of two functions.
To solve this question, we need to understand how the inverse of a function works.
The inverse of a function is defined thus:
[tex]\begin{gathered} \text{if }f(x)=y \\ x=f^{-1}(y) \end{gathered}[/tex]With this definition, we can solve the questions.
Question 1:
[tex]\begin{gathered} f(3)=2 \\ \therefore f^{-1}(2)=3_{} \end{gathered}[/tex]Question 2:
[tex]\begin{gathered} f^{-1}(-2)=-1_{} \\ \\ \therefore f(-1)=-2 \end{gathered}[/tex]Thus, the answers are
[tex]\begin{gathered} \text{Question 1:} \\ f^{-1}(2)=3 \\ \\ \text{Question 2:} \\ f(-1)=-2 \end{gathered}[/tex]
Solve the right triangle with a=60.6 and C= 90 degrees. Round off the results according to the table below
SOLUTION
Given the question in the image, the following are the soluton steps to answer the question.
STEP 1: Draw the given triangle
STEP 2: Write the given parameters
[tex]\theta=90^{\circ},a=60.6[/tex]Since we do not know the measure of any of the two remaining angles, we cannot solve for the required sides and angles.
Hence, There is no enough information to solve the triangle
Is the expression 4sr2(2rs + 3s) completely factored? Complete the sentence with the correct explanation.
The expression 4sr²(2rs + 3s) is not completely factored.
How to factor an expression?An algebraic expression consists of unknown variables, numbers and arithmetic operators.
In other words, an expression or algebraic expression is any mathematical statement which consists of numbers, variables and an arithmetic operation between them.
An expression is completely factored when no further factoring is possible.
Therefore, let's check if the expression is completely factored.
4sr²(2rs + 3s)
The expression still have a common factor which is s. This means its not completely factored.
The complete factorisation is as follows;
4sr²(2rs + 3s) = 4s²r²(2r + 3)
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