we have the function
f(x)=3x^2-54x+241
this is a vertical parabola open upward (because the leading coefficient is positive)
the vertex is a minimum
Convert the quadratic equation into vertex form
y=a(x-h)^2+k
where
(h,k) is the vertex
and the axis of symmetry is equal to the x-coordinate of the vertex
so
x=h
step 1
Factor the leading coefficient
[tex]\begin{gathered} f(x)=3(x^2-18x)+241 \\ \text{complete the squares} \\ f(x)=3(x^2-18x+9^2-9^2)+241 \\ f(x)=3(x^2-18x+9^2)+241-(9^2)\cdot(3) \\ f(x)=3(x^2-18x+81)+241-243 \\ \text{rewrite as p}\operatorname{erf}ect\text{ squares} \\ f(x)=3(x-9)^2-2 \end{gathered}[/tex]the vertex is the point (9,-2)the axis of symmetry is x=9the vertex represent a minimumWhich of these situations can be represented with an integer that, when combined with the opposite of 8, makes 0?
Let A = {a,b,c), B = {4,5,6), and/= {(a ,6), (b,4), (c,6)). Is f a function from A to B? Explain.
The condition for a group of paired values to be a function is that for each value of the independent variable exists one and only one value of the dependant variable.
In this case, A is the group of points that represent the independent variable, and B, the dependent variable.
As each value of A has one and only one value of B, f is a function from A to B.
The inverse is not true, as 6 would have 2 values (a and c) in the image (A).
A is the domain of the independent variable.
f is the function that transform the values present in the group A in some of the elements of the image, that is group B.
The conditions for that relation to be a function is that each of the values in the domain has one and only one value in the image.
A radioactive substance decays so that after t years, the amount remaining, expressed as a percent of the original amount, is A(t)=100(1.3)^-tWhat is the Half-Life of this substance?
Explanation:
The dunction is given below as
[tex]A(t)=100(1.3)^{-t}[/tex]To figure out the half life,
The new mass of the decayed substance will be half that of the initial substance
[tex]\begin{gathered} A(t)=100\left(1.3\right)^{-t} \\ A(t)=\frac{100}{2}=50 \end{gathered}[/tex]By substituting the values, we will have
[tex]\begin{gathered} A(t)=100(1.3)^{-t} \\ 50=100(1.3)^{-t} \\ \frac{50}{100}=1.3^{-t} \\ \frac{1}{2}=1.3^{-t} \end{gathered}[/tex]Apply exponent rules
[tex]\begin{gathered} \frac{1}{2}=1.3^{-t} \\ ln(\frac{1}{2})=-tln(1.3) \\ t=\frac{ln(2)}{ln(1.3)} \\ t=2.64years \end{gathered}[/tex]Hence,
The final answer is
[tex]2.64years[/tex]3. Jill made 20 muffins. She put them into 3 boxes and has two muffins left. How many are in
each box if they all contain the same amount of muffins?
Answer: 6
Step-by-step explanation:
20 (amount of muffins) - 2 (muffins left) = 18 (total in boxes)
18 ÷ 3 = 6
A truck driver travels at an average speed of 65 miles per hour on a 230-mile trip to pick up a load of freight. On the return
55 miles per hour. What is the average speed for the round trip? (Round your answer to one decimal place
Answer:
60.0 miles per hour.
Step-by-step explanation:
The one-way distance is the same as the return distance, so, the average speed is obtained by averaging both speeds:
(65+55)/2 = 120/2 = 60 miles per hour
Use I = PRT to solve.
(time is in years)
| = $14,400
R = 8%
T = 30 years
Find P.
Principal is $6000.
Define simple interest.Simple Interest (S.I.) is a way for figuring out how much interest will accrue on a specific principal sum of money at a certain rate of interest. For instance, if someone borrows $5000 at a rate of 10 p.a. for two years, they will pay S.I. on the borrowed money for those two years. Simple interest is a quick and simple formula for figuring out how much interest will be charged on a loan. The daily interest rate, the principle, and the number of days between payments are multiplied to calculate simple interest.
Given Data
Using simple interest formula,
| = $14,400
R = 8%
T = 30 years
I = [tex]\frac{PRT}{100}[/tex]
14400 = [tex]\frac{P(8)(30)}{100}[/tex]
[tex]\frac{14400(100)}{8(30)}[/tex] = P
P = 6000
Principal is $6000.
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To the nearest tenth of a second, how much time would it take the penny to hit the ground?
In this problem, we want to approximate when a penny will hit the grownd after it has been dropped from the back of the bleachers.
Since we have been given a table, we can make a close estimate based on the y-values.
When the penny hits the ground, the value of y will be 0.
Notice on the table provided that the penny is going to reach 0 between the values of 0.6 and 0.7 seconds.
At 0.6 seconds, it has a distance of 0.236 meters from the ground.
At 0.7 seconds, supposing the penny went past hitting the ground, it was have a distance of
[tex]|-0.401|=0.401\text{ meters from the ground.}[/tex]Since the penny is closer to 0 at the 0.6 s mark than the 0.7 s mark, we can say 0.6 seconds is when the penny will hit the ground.
Here is a graphic model of the penny moving toward the ground:
Which is less (_25)+12 or 40 +(_63)
The answer is 40 + (- 63). Addition is a method of combining items and counting them as a single large group. In mathematics, addition is the process of combining two or more numbers.
What is meant by equation?In its most basic form, an equation is a mathematical statement that shows that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign.
A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation.
Therefore,
First calculate (- 25) + 12(- 25) + 12 = - 13
Calculate 40 + (-63)40 + (- 63) = - 23
we get - 13 and -23 ,
-23 is lesser than -13
-23 < -13.
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If you have a second credit card, your security deposit or spending limit prevents you from spending more than you can afford or repay
True or False?
Answer: true
Step-by-step explanation:
Jarrett surveyed elementary school students about their favorite animals at the zoo He displayed his results in this circle graph If 412 students chose lions as their favorite zoo animal how many total students did Jarrett survey?
Since the 40% of the students choose lion, we can use the following expression:
[tex]\begin{gathered} \text{Total of students}\cdot\frac{\text{percent}}{100}=\text{equivalent number to the percent} \\ \text{Total students}\cdot\frac{40}{100}=412 \\ \text{Isolating the variable total students:} \\ \text{Total students=}\frac{412\cdot100}{40}=\frac{41200}{40} \\ \text{Total students=1030} \end{gathered}[/tex]Jarrett survey 1030 students
Classify the system (options: consistent, independent consistent, inconsistent) 4x + 2y = 8y = -2x + 8
Write both equations in standard form:
[tex]ax+by=c[/tex]The first one is already written in standard form:
[tex]4x+2y=8[/tex]To write the second one in standard form, add 2x to both members of the equation:
[tex]\begin{gathered} y=-2x+8 \\ \Rightarrow y+2x=-2x+8+2x \\ \Rightarrow2x+y=8 \end{gathered}[/tex]Notice that if we divide both members of the first equation by 2, we get:
[tex]\begin{gathered} \frac{4x+2y}{2}=\frac{8}{2} \\ \Rightarrow2x+y=4 \end{gathered}[/tex]Then, the system of equations is equivalent to:
[tex]\begin{gathered} 2x+y=8 \\ 2x+y=4 \end{gathered}[/tex]For the transitive property of equality, since both left members are the same, we can conclude from that system of equations that 8=4, which is false. Then, the system is inconsistent, since it leads to a contradiction.
Therefore, the answer is:
[tex]\text{Inconsistent}[/tex]1. Liezl bought 2 1/4 kilograms of pork, 5 1/2 kilograms of chicken and 1 1/3 kilograms of beef. How many kilograms of meat did she buy in all?
2. Roel travelled 1 1/2 km while Arnel 1 1/2 km more than the distance covered by Roel. How many kilometers did the two travel together?
3. Kenneth and Ian have 20 kilograms of fruits. Kenneth has 9 5/8 kilograms of fruits. How many kilograms of fruits does Ian have?
4. Father has a 17 3/7 m of nylon string. He gave 10 m of it for his brother’s fishing net. How long is the nylon string that is left?
5. Althea is working in a bakeshop. She put two cheesecakes on display. The first cheesecake was cut into 8 slices and there are 5 slices left. The other was cut into 12 slices and there are 4 slices left. How much cheesecake is left.
1. Liezl bought the total amount of meat = 118 / 12 kilograms.
2. Roel travel 3 kilometer.
3. Lan have 83/8 kilograms fruits.
4. Her father left 52 / 7 nylon.
5. The left cheesecake is 9 slices.
What is Fraction?
A fraction is a part of whole number, and a way to split up a number into equal parts.
Now,
1. For first condition;
Amount of pork = 2 1/4 kilograms
Amount of chicken = 5 1/2 kilograms
Amount of beef = 1 1/3 kilograms
So, Total meat = 2 1/4 + 5 1/2 + 1 1/3
= 9/4 + 11/2 + 4/3
= 36/12 + 66/12 + 16/12
= 118 / 12
Thus, The total amount of meat = 118 / 12 kilograms.
2. Total distance cover by Roel = 1 1/2 + 1 1/2
= 3/2 + 3/2
= 6/2
= 3 kilometer.
So, Roel travel 3 kilometer.
3. Kenneth and Lan have fruits = 20 kilograms.
Kenneth have fruits = 9 5/8 kilograms.
So, Amount of fruits does Lan have = 20 - 9 5/8
= 20 - 77/8
= (160 - 77) / 8
= 83 / 8 kilograms.
Thus, Lan have 83/8 kilograms fruits.
4. Father have nylon string = 17 3/7 m
And, He gave 10 m of it for his brother’s fishing net.
So, Left amount of the nylon string = 17 3/7 - 10
= 122/7 - 10
= 122 - 70 / 7
= 52 / 7
Thus, He left 52 / 7 nylon.
5. Since, Althea have left in first cheesecakes = 5 slices
And, In second cheesecakes = 4 slices.
So, Total left cheesecake = 5 + 4
= 9 slices.
Thus, The cheesecake is 9 slices.
Therefore,
1. Liezl bought the total amount of meat = 118 / 12 kilograms.
2. Roel travel 3 kilometer.
3. Lan have 83/8 kilograms fruits.
4. Her father left 52 / 7 nylon.
5. The left cheesecake is 9 slices.
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The statements below represent some possible deposits and withdrawals from Rachel's bank account. Match each statement with the total amount that Rachel's account would change. -47.50 -22.50 22.50 She deposits $35 to her account. Then spends $12.50 from her account on lunch. She withdraws $35 from her account. Then spends $12.50 from her account on lunch. She spends $35 from her account on clothes. Then she deposits $12.50 to her account. 000 OO
merlyumanzor23, this is the solution:
A. 35 - 12.50 =
help meeeeeeee pleaseee
thank you
Part a: The equation is y=-500x+3000
Part b: The prediction of sales is 2750 at a price of $6.5
The equation of a line is given by:
y=mx+b
In which
m is the slope, representing the rate of change.
b is the y-intercept, which is the value of y when x = 0.
Part a:
In this problem:
There are two points (6, 3000) and (8, 2000).
The slope is given by: Change in y/Change in x
m = 2000-3000/8-6
m = -500
Thus
y = -500x+b
Point (6,3000) means that when x=6, y=3000, which we use to find b.
3000 = -500(6)+b
3000=-3000+b
b = 6000
So, y = -500x+6000
Part b:
The sales are y when x = 6.5, thus:
y = -500(6.5)+6000
= -3250+6000
The prediction is 2750 sales with a price of $6.50.
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Given f(x)= -5x^5 - x^4 + 4x^3 and g(x)= -5x^5 - 3x^3 -5x^2 +8 find and simplify f(x) - g(x).
f(x) - g(x) =
degree=
leading coefficient=
The simplified function, f(x) - g(x) is - x⁴ + 7x³ + 5x² - 8.
The degree and leading coefficient are 4 and -1 respectively.
How to simplify functions?The functions to be simplified is as follows;
f(x) = - 5x⁵ - x⁴ + 4x³
g(x) = - 5x⁵ - 3x³ -5x² + 8
Therefore, let's simplify the function,
f(x) - g(x) = - 5x⁵ - x⁴ + 4x³ - (- 5x⁵ - 3x³ -5x² + 8)
open the bracket with the negative sign
f(x) - g(x) = - 5x⁵ - x⁴ + 4x³ + 5x⁵ + 3x³ + 5x² - 8
combine like terms
f(x) - g(x) = - 5x⁵ + 5x⁵ - x⁴ + 4x³ + 3x³ + 5x² - 8
f(x) - g(x) = - x⁴ + 7x³ + 5x² - 8
Therefore,
degree of the function = 4
Leading coefficient = -1
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Which is a polar form of the following parametric equations?x=5cos² thetay = 5 cos theta sin theta
ANSWER
[tex]r(\theta)=5cos\theta[/tex]EXPLANATION
Given;
[tex]\begin{gathered} x=5cos^2\theta \\ y=5cos\theta\:sin\theta \end{gathered}[/tex]Convert from polar form to parametric form.
[tex]\begin{gathered} x(\theta)=r(\theta)cos\theta \\ y(\theta)=r(\theta)sin\theta \end{gathered}[/tex]Here;
[tex]\begin{gathered} x(\theta)=5(cos\theta)^2=5cos\theta\times cos\theta \\ y(\theta)=5cos\theta s\imaginaryI n\theta=5cos\theta sin\theta \\ \Rightarrow r(\theta)=5cos\theta \end{gathered}[/tex]what is 1,000 / 10,000
Answer:
Step-by-step explanation:
Ok this will go into decimals:
1,000/10,000: First delete the zeros as a simple way to solve these problems
SO:
1/10=0.1 as answer
CuIU/u
Rebecca is playing a video game that involves planting and cutting down trees. She
moves up an energy level for each tree she plants, and she moves down a level for
each tree she cuts. If Rebecca reaches energy level 10, she will earn bonus points.
Rebecca is currently at energy level 4. The results of her next 3 turns are:
Turn 1: 6 trees cut
Turn 2: 8 trees planted
Turn 3: tree cut
Answer the questions to find out whether Rebecca earned bonus poirlts in the next 3
turns. Use the number line to help you add.
2 3 4 5 6 7 8 9 10
1. Using only addition, write an expression that represents Rebecca's score after these
three turns. (Remember that she starts at an energy level of 4.). (2 points)
By the end of her three turns, she is level 5.
The number line is what?
A number line is a visual depiction of numbers on a straight line in mathematics. A number line's numerals are arranged in a sequential manner at equal intervals along its length. It is often displayed horizontally and can extend indefinitely in any direction.
If Rebecca cuts down six trees on her next round, she drops down to level four and gains six levels. Rewriting this as -6 yields 4 + (-6) = -2. her following turn, she will plant 8 trees, increasing her level by 8. -2 + 8 = 6 In her final round, she removes one tree.
6 + (-1) = 5
4 + (-6) + 8 + (-1) = 5 in total.
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help me please
thank you
Answer:
Domain: [tex)(-\infty, \infty)[/tex]
Range: [tex][0, \infty)[/tex]
Step-by-step explanation:
The domain is the set of x-values and the range is the set of y-values.
What is the value of 6x exponent 4 when x = 5
Answer:
6480
Step-by-step explanation:
6^4x5
Answer:
3750
Step-by-step explanation:
Hey there!
To solve this we must first make this into an equation
[tex]6x^4\\x=5\\6(5)^4\\6(625)\\3750[/tex]
h(x) is given. h(x)=f(g(x)). Give a possible combination of f(x) and g(x).
[tex]h(x)=(x-4)^3+(x-4)^\frac{3}{2} -1[/tex]
IN THEVPLACE OF x in the EQUATION OF f(x) we find that x-4 is plugged in..
THIS brings us to a result that g(x) is accompanied by x-4
[tex]g(x) = x - 4[/tex]
and For f(x) it shows
[tex]f(x) = {x}^{3} + {x}^{ \frac{3}{2} } - 1[/tex]
SINCE THE POSITION OF THE X VALUES OF WHICH x-4 was plugged in has to be reserved.
PLEASE IF YOU DON'T UNDERSTAND MESSAGE ME QUICKLY.
Solve the inequality for u.u +8 > 24Simplify your answer as much as possible.DDDD
Answer:
[tex]u\text{ }>\text{ 16}[/tex]Explanation:
Here, we want to solve the given inequality for u
What we have to do is to bring like terms together
We bring 8 to the right which causes a sign change
Mathematically, we have that as:
[tex]\begin{gathered} u\text{ }>\text{ 24-8} \\ u\text{ }>\text{ 16} \end{gathered}[/tex]((1/3)x^2))-((1/9)x)-(2/9)=0
The solution of the quadratic equation (1/3)x² - (1/9)x - 2/9 = 0 are x = 1, - 2/3.
Any equation that can be written in the standard form where x stands for an unknown and a, b, and c are known quantities, with a 0, is said to be quadratic.
An assignment of values to the unknown variables that establish the equality in the equation is referred to as a solution.
Consider the expression,
(1/3)x² - (1/9)x - 2/9 = 0
Multiplying each side of the quadratic equation by 9,
9 × [ (1/3)x² - (1/9)x - 2/9 = 0 ]
3x² - x - 2 = 0
Using the quadratic formula,
[tex]x = \frac{-b \pm \sqrt{b^2-4ac} }{2a}[/tex]
[tex]x = \frac{1 \pm \sqrt{1+24} }{6}\\ x = \frac{1 \pm \sqrt{25} }{6}[/tex]
x = 1 and x = - 2/3
Hence, the solution of quadratic equation (1/3)x² - (1/9)x - 2/9 = 0 are x = 1, - 2/3.
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Is 5 a solution to 3x-8=0?
The equation is 3x - 8 = 0.
So, 5 is a solution if we replace x by 5 and we get 0 on the left side of the equation.
Then, replacing x by 5, we get:
3x - 8 = 0
3*5 - 8 = 0
15 - 8 = 0
7 = 0
Since 7 is different from 0, 5 is not a solution.
Answer: 5 is not a solution
The dance committee hired a DJ for the fall dance. The DJ charges $125 per hour plus $75 for an assistant. The committee wants to keep the total cost under $450. plssss I need stepss.
The dance committee can hire the DJ for a maximum of 3 hours keeping the cost under control
DJ charges per hour = $125
Charges for assistance = $75
Budget by the committee = $450
Mathematical expressions with inequalities are those in which the two sides are not equal. Contrary to equations, we compare two values in inequality. Less than (or less than or equal to), greater than (or greater than or equal to), or not equal to signs are used in place of the equal sign.
Let x represent the maximum number of hours to hire the DJ
Formulating the equation we get:
DJ charges per hour*Number of hours<= Required budget
125x + 75<=450
Solving the equation:
125x <= 375
x <=3
So, a maximum of 3 hours
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Match each equation to one of the two tape diagrams. Dragon drop each card to match the equation to the correct tape diagram. - (please use ss)
First diagram:
Sum of the three parts is equal to 9: x+x+x is 9 or three times x is 9
Equations:
[tex]\begin{gathered} 3\cdot x=9 \\ 9=3\cdot x \\ x+x+x=9 \end{gathered}[/tex]Also you can write the equation 3*x=9 as follow by solving x:
[tex]x=9\div3[/tex]Second diagram:
Sum of the two parts is equal to 9: x+3 = 9
Equations:
[tex]\begin{gathered} x+3=9 \\ 3+x=9 \end{gathered}[/tex]Solving x you can write the equation:
[tex]x=9-3[/tex]The current temperature is 48°F. It is expected to drop 1.5°F each hour.
The current temperature is 48°F, in expected to drop 1.5°F each hour, then the hours per word 8 hours the temperature will be 36°F.
The current temperature is = 48°F.
It is expected to drop = 1.5°F each hour.
We can assume, temperature = 36°F
Let x hours the temperature will be 36°F.
Temperature is a measure of the average kinetic energy of the particles in an object. When the temperature increases, the motion of these particles also increases. Temperature is measured with a thermometer or a calorimeter.
Then,
48 - 1.5x = 36
It's expected to drop 1.5F per hour,
We can list the equation,
So,
We can write,
48 - 1.5x = 36
48 - 36 = 1.5x
We can sum or difference of the product,
1.5x = 12
x = 12/1.5
We can divide the products,
x = 8 hours
Therefore,
In expected to drop 1.5°F each hour, then the hours per word 8 hours the temperature will be 36°F.
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Answer:
The present temperature is 48°F; it is anticipated to decrease by 1.5°F every hour, reaching 36°F in 8 hours.
Step-by-step explanation:
48°F is the current temperature.
An hourly decrease of 1.5°F is anticipated.
We'll assume that it's 36°F.
The temperature will be 36°F in x hours.
The average kinetic energy of the particles in an object is measured by its temperature. The velocity of these particles likewise increases as the temperature rises. A thermometer or a calorimeter is used to determine the temperature.
Then,
48 - 1.5x = 36
It's expected to drop 1.5F per hour,
We can list the equation,
So,
We can write,
48 - 1.5x = 36
48 - 36 = 1.5x
We can sum or difference of the product,
1.5x = 12
x = 12/1.5
We can divide the products,
x = 8 hours
Therefore,
In expected to drop 1.5°F each hour, then the hours per word 8 hours the temperature will be 36°F.
Write an equation and point slope for the line that passes through the giving order pairs there are two correct answers for this question Depending upon which order pair do you use for the slope for
If we label the given points to find the slope of the line, we have:
(4, 1 ) ---> x1 = 4, y1 = 1
(-2, 7) ---> x2 = -2, y2 = 7.
Then, using the formula for the slope of a line, we have:
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{7-1}{-2-4}=\frac{6}{-6}\Rightarrow m=-1[/tex]Then, having this value of m, we can apply the formula for the Point-Slope Form of the line:
1. In the case we take the pair as ( 4, 1), the equation is:
[tex]y-y_1=m(x-x_1)=y-1=-1(x-4)=y-1=-(x-4)[/tex]2. In the case we take the pair as (-2, 7), we have:
[tex]y-7=-1(x-(-2)=y-7=-(x+2)[/tex]Then, we have two options:
[tex]y-1=-(x-4)[/tex][tex]y-7=-(x+2)[/tex]1, 728 ÷ 32 what is the value of the expression
Result : 54
reminder = 0
[tex]1,728\div32[/tex]
Simplify the expression:
[tex]1,728\div32\\[/tex]
[tex]=54[/tex]
Hope this helps!
The function shown in the graph is f(x) =
Answer:
x³-2x²-5x+6 =0
Step-by-step explanation:
x³-2x²-5x+6 =0
checking zeros
x=1 , 1-2-5+6=0, 0=0 true
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