Given:
[tex]6^{9\text{ }}\ast6^x=6^2[/tex]To find the value of x, first apply exponential property which is:
[tex]a^m\text{ }\ast a^{n\text{ }}=a^{m+n}[/tex]Now we have:
[tex]6^{9+x\text{ }}=6^2[/tex]Since both bases are equal, let's remove both bases, take the exponent and find x:
[tex]9\text{ + x = 2}[/tex]Now subtract from both sides:
[tex]9\text{ - 9 + x = 2 - }9[/tex][tex]0\text{ + x }=\text{ -7}[/tex][tex]x\text{ = -7}[/tex]The value of x is -7
1. Which of the following have the same domain and range? I. y = 2x - 1 II. y = -x + 2 x = 2 (A) I and III only (B) I and II only (C) II and III only (D) I, II and III
The I and II are linear function, so those have the same domain and the same range always. So the answer is B
Pat is walking to a restaurant 9 blocks away. Pat has walked 3 blocks in 7minutes. How many more minutes to reach the restaurant?
We can solve this question in the following way:
We have that Pat has walked 3 blocks in 7 minutes, and now, we can post the following: how long will it take to walk 9 blocks?
Then, we have:
Then, we can solve for x as follows:
We need to multiply 9 blocks by 7 minutes, and then we divide the result by 3 blocks (see the red lines).
[tex]x=\frac{9bl\cdot7\min}{3bl}\Rightarrow x=\frac{63\min}{3}\Rightarrow x=21\min [/tex]This answer comes from having into account the following proportion:
[tex]\frac{9bl}{3bl}=\frac{x}{7\min}\Rightarrow x=\frac{9bl\cdot7\min}{3bl}=21\min [/tex](We have here proportional quantities.)
It will take 21 min to walk the 9 blocks. However, Pat has walked for 7 minutes. Therefore, Pat needs to walk for (21 min - 7 min) = 14 minutes.
Therefore, Pat will need to walk for 14 minutes more to reach the restaurant.
Answer:
15 minutes
Step-by-step explanation:
15 minutes
The blocks remaining is 7 - 2 = 5. The rate is (2 blocks)/(6 min).
So (2 blocks)/(6 min) = (5 blocks)/(x min).
2/6 = 5/x (units omitted for simplicity)
(2/6) × x = (5/x) × x (Multiplying both sides by x so x is not at the bottom).
(2/6) × x = 5
6 × (2/6) × x = 6 × 5
2x = 30
x = 15
Pat will take 15 minutes to walk the remaining 5 blocks.
triangle W quadrilateral hexagon pentagon 2 + please help me out with thisWhat is a name for this shapetrianglequadrilateralhexagonpentagon
To determine the name of the shape you have to count its sides.
If it has 3 sides is a triangle, if it has 4 sides is a quadrilateral, if it has 5 sides is a pentagon and if it has 6 sides is a hexagon,
The shape has 6 sides so it is a hexagon.
n applicant receives a job offer from two different companies. Offer A is a starting salary of $58,000 and a 3% increase for 5 years. Offer B is a starting salary of $56,000 and an increase of $3,000 per year.
Part A.
The inital salary is $58,000, then we have:
[tex]a_1=58000_{}[/tex]Since we have an increase of 3% each year we know that the second year the salary would be:
[tex]\begin{gathered} a_2=1.03a_1 \\ a_2=1.03\cdot58000 \end{gathered}[/tex]The third year the salary would be:
[tex]\begin{gathered} a_3=1.03a_2 \\ a_3=1.03(1.03)58000 \\ a_3=(1.03)^258000 \end{gathered}[/tex]and so on for year 4 and 5.
Since the increase in salary is only the first five years we conclude that this can't be represented by a geometric series.
For the first five year we can calculate the salary using a geometric sequence with common ratio 1.03, then for the first five years the salary is given by
[tex]a_n=(1.03)^{n-1}_{}\cdot58000\text{ for }1\leq n\leq5[/tex]The salary for the any subsequent year is given by:
[tex]a_n=(1.03)^4\cdot58000\text{ for }n>5[/tex]Part B.
Since we are adding a certain quantity each year we conclude that this offer can be represetend by an algebraic series given by:
[tex]\begin{gathered} b_n=56000+(n-1)3000 \\ b_n=56000+3000n-3000 \\ b_n=3000n+53000 \end{gathered}[/tex]Part C.
After five years the income for offer A is:
[tex]a_5=(1.03)^4\cdot58000=65279.51[/tex]For offer B is:
[tex]b_5=3000(5)+53000=68000_{}[/tex]Therefore after 5 years job offer B has a greater total income.
Find the derivative of the trigonometric function using the Chain Rule.[tex]y = \cos \sqrt{x} [/tex]
I need the answer but i also need to check my answer after
we have the following:
[tex]\begin{gathered} 7=\frac{x-8}{3} \\ \end{gathered}[/tex]solving for x:
[tex]\begin{gathered} x-8=7\cdot3 \\ x=21-8 \\ x=13 \end{gathered}[/tex]y= 4/3 x-1 graph the line the top to right is 10 8 6 4 2 and at the left to bottom is -10 -8 -6 -4 -2
the graph of this equation is:
Let us make part of the table corresponding to this graph
how to write this on a number line1 plus x less than 5
We are given the following inequality
[tex]1+x<5[/tex]Let us first solve the inequality for x.
Subtract 1 from both sides of the inequality
[tex]\begin{gathered} -1+1+x<5-1 \\ x<4 \end{gathered}[/tex]So the solution of the inequality is all the values of x less than 4
Now let us graph this solution on a number line.
Hi I need help bad on this I have a report card Wednesday and my birthday is in 13 days
3)
- 18/(- 6)
This can be written as
- 18/- 6
Recall, if a negative number divides a negative number, the result is positive. Thus, the answer is 3
Which number is the same as (4-1)20-4-2), A - 2 B. 1/8C 2 D. 32 E. 512
Which equation is the inverse of y = 16x2 + 1?O y=+T6x-1O y =16+Oy AVTOINX-1Оy=
We need to find the inverse function of:
[tex]undefined[/tex]A hose for the hot tub at a rate of 3.61 gallons per minute. How many hours will it take to fill a 345 gallon hot tub?
We have a 345 gallon hot tub that fills at a rate of 3.61 gallons/min.
To calculate filling time, we need to divide as it follows:
[tex]t=\frac{345\text{gal}}{3.61\frac{\text{gal}}{\min }}[/tex]in that way we can find how much time does it take in minutes
[tex]t=\frac{345}{3.61}=95.56786[/tex]now, if we want to determine time in hours, we must divide by 60, because there are 60 min. in one hour.
[tex]t=\frac{345}{3.61}\cdot\frac{1}{60}=1.59279[/tex]Then, it would take approximately 1.59 hours to fill the hot tub.
Function F, shown below, assigns to a temperature given in degrees Celsius it's equivalent in degrees Fahrenheit. Function C, also shown below, assigns to a temperature given in degrees Kelvin its equivalent in degrees Celsius. Choose the response below that shows the correct expression for F(C(x)) and then choose the response below that correctly interprets the meaning of F(C(x)).F(x)=(9/5)x+32 C(x)=x−273
Given:
The expressions are given as,
[tex]\begin{gathered} F(x_)\text{ = \lparen}\frac{9}{5})x\text{ + 32} \\ C(x)\text{ = x - 273} \end{gathered}[/tex]Required:
The value of F(C(x)) and its response.
Explanation:
The required function is calculated as,
[tex]\begin{gathered} F(C(x))\text{ = }\frac{9}{5}\text{ \lparen x - 273\rparen + 32} \\ F(C(x))\text{ = }\frac{9x}{5}\text{ - }\frac{2457}{5}\text{ + 32} \\ F(C(x))\text{ = }\frac{9x}{5}\text{ + 32 - }\frac{2457}{5} \\ F(C(x))\text{ = }\frac{9x}{5}\text{ + }\frac{160}{5}\text{ - }\frac{2457}{5} \\ F(C(x))\text{ = }\frac{9x}{5}\text{ - }\frac{2297}{5} \\ F(C(x))\text{ = }\frac{9x}{5}\text{ - 459}\frac{2}{5} \\ \end{gathered}[/tex]Questionf(2)Find R(2) where f(2)g(2)x² – a- 2 - 3010x + 100and g(2)-22 – 5x + 6611x + 110(Simplify your answer.)Provide your answer below:
Answer:
Given that,
To find,
[tex]R(x)=\frac{f(x)}{g(x)}[/tex]where,
[tex]f(x)=\frac{x^2-x-30}{10x+100}[/tex][tex]g(x)=\frac{-x^2-5x+66}{11x+110}[/tex]Simplifing f(x) and g(x), we get
[tex]f(x)=\frac{x^2-x-30}{10x+10)}=\frac{x^2-6x+5x-30}{10(x+10)}[/tex][tex]=\frac{x(x-6)+5(x-6)}{10(x+10)}=\frac{(x-6)(x+5)}{10(x+10)}[/tex][tex]f(x)=\frac{(x-6)(x+5)}{10(x+10)}-----(1)[/tex]This is the simplified form of f(x).
For g(x) we get,
[tex]g(x)=\frac{-x^2-5x+66}{11x+110}=\frac{x^2+5x-66}{-11(x+10)}[/tex][tex]=\frac{x^2+11x-6x-66}{-11(x+10)}=\frac{x(x+11)-6(x+11)}{-11(x+10)}[/tex][tex]g(x)=\frac{(x+11)(x-6)}{-11(x+10)}------(2)[/tex][tex]\frac{i}{g(x)}=\frac{-11(x+10)}{(x+11)(x-6)}[/tex]Now To find R(x), we get
[tex]R(x)=\frac{f(x)}{g(x)}=f(x)\times\frac{1}{g(x)}[/tex][tex]=\frac{(x-6)(x+5)}{10(x+10)}\times\frac{-11(x+10)}{(x+11)(x-6)}[/tex][tex]=\frac{-11(x+5)}{10(x+11)}[/tex]we get,
[tex]R(x)=\frac{-11(x+5)}{10(x+11)}[/tex]Answer is:
[tex]R(x)=\frac{-11(x+5)}{10(x+11)}[/tex]Complete the ratio table of the median price of renting a two-bedroom apartment by finding the value of x and y. Round answers to two decimal places. Norfolk $951 Richmond $1,042 1 X Х 100 у To solve the values set up and solve a a. Bar Chart b. Ratio c. Proportion d. Weighted Average x = y =
2+4=6 is true
7*8=56 is true
So the statement 2+4=6 AND 7*8=56 is also true
So the answer for question #36 is a) the statement is true because both proporsitions are true
please help I don't understand how to find the volume of the cylinder(please add an explanation).
Explanation
From the image, we can see that the radius of the cylinder is given as
[tex]\frac{d}{2}=\frac{20}{2}=10ft[/tex]The height of the cylinder is 40ft. Therefore, the volume of the cylinder is
[tex]volume=\pi r^2h=3.14\times10^2\times40=12560[/tex]Answer: 12560 cubic feet
Find the coordinates of the missing vertex of rectangle ABCD with A(-3, 3),B(5, 3), and D(-3, -1).O (5, -1)© (-11, 7)O (5,3)O (1, -1)
Given:
The given vertex of a rectangle ABCD are A=(-3,3), B=(5,3) and D=(-3,-1)
To find: Missing vertex, that means vertex C
The graph is as follows:
From the above graph, the coordinates of point C are (5,-1).
Hence, the required answer is (5,-1).
Question 3 Olivia is making pancakes for breakfast. The recipe calls for 0.5 quart of milk and 2.5 cups of flour. She has quart of 3/8 quart of milk and 18/8 cups of flour. Olivia makes the recipe with the milk and flour she has. Explain her error. Hint: Convert all of them to either decimal or fraction so that you can compare the values. Challenge: How much more or less milk does Olivia need? How much more or less flour does Olivia need?
ANSWER and EXPLANATION
The recipe calls for 0.5 quart of milk and 2.5 cups of flour.
She has 3/8 quart of milk and 18/8 cups of flour.
To know the error she made, we have to find the ratio of milk to flour in the recipe and the ratio of milk to flour that she used and see if the ratios are the same.
RECIPE
Ratio of milk to flour is:
[tex]\begin{gathered} 0.5\text{ : 2.5} \\ \Rightarrow\text{ 1 : 5} \end{gathered}[/tex]That is the ratio of milk to flour.
USED BY OLIVIA
Ratio of milk and flour that she used is:
[tex]\begin{gathered} \frac{3}{8}\text{ : }\frac{18}{8} \\ =>\text{ 3 : 18} \\ \Rightarrow\text{ 1 : 6} \end{gathered}[/tex]We can already see that the ratios are not the same.
By comparing the ratios, we see that she actually used more flour than she was supposed to.
which is the best description of the data in the scatter plot
Types of correlation:
Based on the types of correlation, we can see that the best description of the data given is a positive correlation.
Answer: A positive correlation.
Write the standard form of the quadratic function f(x) whose graph has vertex (1,2) and passes through (2,4)
Step 1. We are given the vertex of the quadratic function:
[tex](1,2)[/tex]And a point:
[tex](2,4)[/tex]Required: Find the standard form of the quadratic equation.
Step 2. Since we know the vertex of the quadratic function we will start by using the vertex form of the quadratic function:
[tex]y=a(x-h)^2+k[/tex]Where (h, k) is the vertex, in this case:
[tex]\begin{gathered} h=1 \\ k=2 \end{gathered}[/tex]Step 3. To use the previous equation
[tex]y=a(x-h)^{2}+k[/tex]We will need to find the value of a. For that, we substitute the h and k values:
[tex]y=a(x-1)^2+2[/tex]And as the values of x and y, we substitute the values of the given point (2,4) where x=2 and y=4
[tex]4=a(2-1)^2+2[/tex]Solving for a:
[tex]\begin{gathered} 4-2=a(1)^2 \\ 2=a(1) \\ 2=a \end{gathered}[/tex]Step 4. Now that we know that the value of a is 2, we go back to our general equation:
[tex]y=a(x-h)^{2}+k[/tex]Substitute the value of a, h, and k:
[tex]y=2(x-1)^2+2[/tex]This is the equation in the vertex form, but we need it in standard form.
Step 5. The standard form of the quadratic function is:
[tex]f(x)=ax^2+bx+c[/tex]To convert our equation into the standard form, first, we change y by f(x):
[tex]\begin{gathered} y=2(x-1)^{2}+2 \\ \downarrow \\ f(x)=2(x-1)^2+2 \end{gathered}[/tex]Then, we use this formula for the binomial squared:
[tex](a-b)^2=a^2-2ab+b^2[/tex]The result is:
[tex]f(x)=2(x^2-2x+1)+2[/tex]Simplifying:
[tex]\begin{gathered} f(x)=2x^2-4x+2+2 \\ \downarrow \\ \boxed{f\mleft(x\mright)=2x^2-4x+4} \end{gathered}[/tex]That is the standard form of the quadratic function.
Answer:
[tex]\boxed{f(x)=2x^{2}-4x+4}[/tex]11.Solve the given equation over the interval [0, 271): 2 sin x++3 = 0.117and x=64757X=- and x=332лx= -- and x=3T57x= -- and x=6and x3
Question:
Solution:
Let the following trigonometric equation:
[tex]2\sin (x)+\sqrt[]{3}=\text{ 0}[/tex]Subtract the root of 3 from both sides of the equation:
[tex]2\sin (x)=-\sqrt[]{3}[/tex]solve for sin(x):
[tex]\sin (x)=\text{ -}\frac{\sqrt[]{3}}{2}[/tex]Applying the trigonometric circle on the given interval, we obtain that the correct answer is:
[tex]x\text{ = }\frac{4\pi}{3}\text{ , x = }\frac{5\pi}{3}[/tex]
Should Jenna buy the smart phone at top quality or big value? support your answer with a mathematical evidence. Assume that getting the lowest price is Jenna's only consideration.?
Let the price of the item be t as stated in the question.
This means if Top quality is selling them at 15% off the list price, then the new price can be represented as;
(A)
[tex]\begin{gathered} Price=t-discount \\ \text{Price}=t-(t\times0.15) \\ \text{Price}=t-0.15t---(1) \\ \text{The second expression that can be used to }represent\text{ the discounted price is;} \\ \text{Price}=(t-0.15t) \\ \text{Price}=0.85t---(2) \end{gathered}[/tex](B)
Equation 1 shows the original list price less the discounted amount (which is 15 percent times the list price, t). The result is the price now paid eventually
Equation 2 shows the percentage of the list price that would be paid by Jenna after deducting the discount, which means she would be paying 85 percent of the list price (that is 0.85)
(C)
A smartphone on sale at 1/4 off its list price, would also mean its being sold at a discount of 25%. One-quarters of 100 percent would be 25, hence the smartphone is at 25% off the list price.
However, where the phone is being sold at 75% of its list price means, the list price now has 25% taken off. That is, the price at Big Value is
[tex]\begin{gathered} \text{Price}=0.75t \\ \text{Discount=t-0.75t} \\ \text{Discount}=0.25 \end{gathered}[/tex]That means the discount at Big value is 25% (or 0.25)
The discount at Top Quality is 25% (0.25 or 1/4)
Jenna can buy at either of the store. Since she is already at Top Quality, she can just go ahead and buy it right there
What is the multiplicity of each of the roots of this graph?2
SOLUTIONS
What is the multiplicity of each of the roots of this graph?
[tex]f(x)=2x^4+12x^3+16x^2-12x-18[/tex]Factorise f(x) by the options
(a) According to the option we have -3 is a root of so x+3 is a factor
[tex]\frac{2x^4+12x^3+16x^2-12x-18}{x+3}=2x^3+6x^2-2x-6[/tex](b) 1 is a root too so x - 1 is a factor
[tex]\frac{2x^3+6x^2-2x-6}{x-1}=2x^2+8x+6[/tex][tex]\begin{gathered} 2x^2+8x+6=2(x+3)(x+1) \\ f(x)=2(x+3)^2(x+1)(x-1) \end{gathered}[/tex]Therefore the correct answer = Option A
Which of the following choices are equivalent to the expression below? Check all that applyx^(3/8)
Given:
[tex]x^{\frac{3}{8}}[/tex]To find:
the equivalence of the given expression
[tex]\begin{gathered} We\text{ will apply exponent rule:} \\ x^{\frac{1}{b}}\text{ = }\sqrt[b]{x} \\ x^{\frac{a}{b}}\text{ = \lparen}\sqrt[b]{x})\placeholder{⬚}^a \\ \\ Applying\text{ same rule to the given expression:} \\ x^{\frac{3}{8}}\text{ = \lparen}\sqrt[8]{x})\placeholder{⬚}^3 \end{gathered}[/tex][tex]\begin{gathered} (\sqrt[8]{x})\placeholder{⬚}^3\text{ can also be written as \lparen}\sqrt[8]{x^3}) \\ x^{\frac{3}{8}}=\text{ \lparen}\sqrt[8]{x^3}\text{ \rparen} \end{gathered}[/tex][tex]\begin{gathered} from\text{ \lparen}\sqrt[8]{x})\placeholder{⬚}^3,\text{ }\sqrt[8]{x}\text{ = x}^{\frac{1}{8}} \\ \\ (\sqrt[8]{x})\placeholder{⬚}^3\text{ = \lparen x}^{\frac{1}{8}})\placeholder{⬚}^3 \\ =(\text{x}^3)\placeholder{⬚}^{\frac{1}{8}} \end{gathered}[/tex]can you help me please
If the value of x increases and the value of y also increases then we can say that the slope is Increases,
If the value of x increases but the value of y decreases then we can say that the slope is decreases.
In the given coordinates of tempreature and icecreams sold is
as the value of Tempreature increases from 60 to 80, the number of icecreams (value of y) is also increases,
but at tempreature =90 the number of icecreams is 20,
thus, the value of y is decreases at x=90
Hence slope changes sign
first it gets positive from tempreature (0 to 80) and then decreases to 90
Answer : C) Slope changes sign
y=-x^2-4x-8Identify the vertex, the axis of symmetry, the maximum or minimum value, and the range of the parabola.
Here we have the following parabola:
[tex]y=-x^2-4x-8[/tex]To find the vertex, we could use the following formula:
[tex]V(x,y)=V(\frac{-b}{2a},\frac{-b^2}{4a}+c)[/tex]Where a, b and c are the coefficients of the quadratic function:
[tex]y=ax^2+bx+c[/tex]As you can see, in this problem a = -1 , b = -4 and c = -8. Thus,
[tex]V(x,y)=V(\frac{-(-4)}{2(-1)},\frac{-(-4)^2}{4(-1)}-8)[/tex]This is:
[tex]V(-2,-4)[/tex]Then, the vertex of the parabola is (-2,-4)
The axis of symmetry of the parabola is the line x=-2. Since the vertex is situated at the coordinates (-2,-4), that means that the parabola is symmetrical around this line.
The vertex is maximum point of the parabola.
The range, is defined as all the values that the y-axis could take. If we notice, that is:
[tex](-\infty,-4\rbrack[/tex]I'm going to upload a picture of the parabola:
please need a answer
Answer:
ok you can ask
Step-by-step explanation:
don't forget to follow rate like
Plot the points (-3,4) and (4,4) on the coordinate plane below.What is the distance between these two points?
To find the distance between two points A and B you can use the formula
[tex]\begin{gathered} d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \text{ Where A and B have the coordinates} \\ A(x_1,y_1) \\ B(x_2,y_2) \end{gathered}[/tex]So, in this case, you have
[tex]\begin{gathered} A(-3,4) \\ B(4,4) \end{gathered}[/tex][tex]\begin{gathered} d=\sqrt[]{(4_{}-(-3))^2+(4-4)^2} \\ d=\sqrt[]{(4_{}+3)^2+(0)^2} \\ d=\sqrt[]{(7)^2} \\ d=7 \end{gathered}[/tex]Therefore, the distance between these points is 7 units.
Graphically,
-3x+5(x+2) find the equivalent
Answer
-3x + 5(x + 2) = 2x + 10
Explanation
The way to find this equivalent is simply to solve the given expression.
-3x + 5(x + 2)
= -3x + 5x + 10
= 2x + 10
Hope this Helps!!!
a garden table and a bench cost $613 combined. the garden table cost $87 less then the bench. what is the cost of the bench ?
the cost of the bench is 350
Explanation
Step 1
let
x represents the cost of the garden table
y represents the cost of the bench
so
a)a garden table and a bench cost $613 combined.
[tex]x+y=613\rightarrow equation\text{ (1)}[/tex]b)the garden table cost $87 less then the bench,( in other words, you have to add 87 to the price of the garden table to obtain the cost of the bech)
[tex]\begin{gathered} x+87=y\rightarrow Equation(2) \\ \end{gathered}[/tex]Step 2
solve the equations
a)isolate x in equaiton (1) and replace in equation (2)
[tex]\begin{gathered} x+y=613\rightarrow equation\text{ (1)} \\ \text{subtrac y in both sides} \\ x+y-y=613-y \\ x=613-y \end{gathered}[/tex]now, replace in equation (2)
[tex]\begin{gathered} x+87=y\rightarrow Equation(2) \\ (613-y)+87=y\rightarrow Equation(2) \\ 700-y=y \\ \text{add y in both sides} \\ 700-y+y=y+y \\ 700=2y \\ \text{divide both sides by 2} \\ \frac{700}{2}=\frac{2y}{2} \\ 350=y \end{gathered}[/tex]it means, the cost of the bench is 350