6x + 1 = 6 + 2x.
collect the like term
6x - 2x = 6-1
4x = 5
divide both-side of the equation by 4
4x/4 = 5/4
x=1.25
1.25
A scale drawing of a game room is shown below:A rectangle is shown. The length of the rectangle is labeled 2 inches. The width of the rectangle is labeled 4.5 inches. The scale is 1 to 30.What is the area of the actual game room in square feet? Round your answer to the nearest whole number.9 ft223 ft256 ft2270 ft
The scale factor from the drawing to the room is 1 to 30. Then, multiply the dimensions of the drawing by 30 to obtain the real dimensions of the room. Then, use the real values to find the area of the room.
Since the length is labeled 2 inches, the real length of the room is:
[tex]2in\times30=60in[/tex]Since the width is labeled 4.5 inches, the real with of the room is:
[tex]4.5in\times30=135in[/tex]1 foot is equal to 12 inches. Then, divide the dimensions by 12 to find the measurements in feet:
[tex]\begin{gathered} 60in=60in\times\frac{1ft}{12in}=5ft \\ \\ 135in=135in\times\frac{1ft}{12in}=11.25ft \end{gathered}[/tex]Multiply the width and the length to find the area of the room:
[tex]A=(5ft)(11.25ft)=56.25ft^2\approx56ft^2[/tex]Therefore, to the nearest whole number, the area of the game room is 56ft^2.
find the value of tan A in simplest radical form
In the given right angle triangle BCA : BC = 5, CA = 3 and BA = root 34
From the trignometric ratio of right angle triangle :
The tangent of angle is the ratio of the Adjacent side to the opposite side
[tex]\tan \theta=\frac{Opposite\text{ side}}{Adjacent\text{Side}}[/tex]In the given triangle, the side opposite to angle A = BC and adjacent side CB
Substitute the value :
[tex]\begin{gathered} \tan \theta=\frac{Opposite\text{ side}}{Adjacent\text{Side}} \\ \tan A=\frac{BC}{CB} \\ \tan A=\frac{5}{3} \\ \tan A=1.66 \\ \\ ^{} \end{gathered}[/tex]The value of tanA = 5/3 or 1.66
Solve the quadratic equation using any algebraic method. Show all work that leads to your answer.
6x² + 23x + 20 = 0
Answer:
x = - [tex]\frac{5}{2}[/tex] , x = - [tex]\frac{4}{3}[/tex]
Step-by-step explanation:
6x² + 23x + 20 = 0 ( factorise the left side )
consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term.
product = 6 × 20 = 120 and sum = 23
the factors are + 8 and + 15
use these factors to split the x- term
6x² + 8x + 15x + 20 = 0 ( factor the first/second and third/fourth terms )
2x(3x + 4) + 5(3x + 4) = 0 ← factor out (3x + 4) from each term
(3x + 4)(2x + 5) = 0
equate each factor to zero and solve for x
2x + 5 = 0 ⇒ 2x = - 5 ⇒ x = - [tex]\frac{5}{2}[/tex]
3x + 4 = 0 ⇒ 3x = - 4 ⇒ x = - [tex]\frac{4}{3}[/tex]
Write a pair of complex numbers whose sum is -4 and whose product is 53
The pair of complex numbers whose sum is -4 and whose product is 53 is illustrated as -b² - 4b - 53 = 0.
How to calculate the he value?Let the numbers be represented as a and b.
Therefore a + b = -4 .....i
a × b = 53 ........... ii
From equation I, a = -4 - b
Put this into equation ii
ab = 53
(-4 - b)b = 53
-b² - 4b = 53
Equate to 0
-b² - 4b - 53 = 0
The value can be found using the Almighty formula
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Which is an x-intercept of the continuous function in thetable?O (-1,0)O (0, -6)O (-6, 0)O (0, -1)
The x-intercept happens when:
[tex]f(x)=0[/tex]Therefore, the x-intercepts for that functions are:
[tex]\begin{gathered} x=-1 \\ x=2 \\ x=3 \end{gathered}[/tex]Answ
Kaitlin got a prepaid debit card with $15 on it. For her first purchase with the card, she bought some bulk ribbon at a craft store. The price of the ribbon was 21
cents per yard. If after that purchase there was $4.92 left on the card, how many yards of ribbon did Kaitlin buy?
yards
Answer:
Kaitlin bought 48 yards of ribbon.
Step-by-step explanation:
Hey! Let's help you with your question here!
So, let's start by figuring out what we know and what we need to figure out. First of all, we started with $15 and ended with $4.92. We also know that the price of the ribbon is $0.21 per yard and we need to figure out how many yards of ribbon she purchased.
In order to figure this out, we first want to know the difference in the price between what we started and what we ended up with. So, we can subtract! It would look like this:
[tex]15-4.92=10.08[/tex]
So, we figured out that the difference in the price is $10.08, but how do we find out how many yards of ribbon Kaitlin bought? Well, since we know that it is $0.21 cents for a yard of ribbon, we can just take the difference in price and divide it by how much a ribbon cost for a yard of it. So it would look like this:
[tex]10.08/0.21=48[/tex]
We have a nice whole number and that's our answer! Therefore, Kaitlin bought 48 yards of ribbon.
Mathematical Way:
To do it in a more mathematical way, we can put it in the form of a formula. We know that the end total is $4.92 and the initial is $15. We also know that it's $0.21 cents per yard of ribbon but we don't know how many yards she bought. We can let the number of yards she bought represent x in the formula, so we have:
[tex]15=0.21x + 4.92[/tex]
This formula makes sense because we start with $15 at the beginning, so we want to add $4.92 from 0.21x because the end total is the remainder of how many yards Kaitlin bought. The process is essentially the same as the method above. If we were to solve the formula, it would give us the same answer:
[tex]15=0.21x+4.92[/tex]
[tex]15-4.92=0.21x[/tex] - Moving the 4.92 over to the left side, beginning to isolate x.
[tex]10.08=0.21x[/tex] - Subtracting $4.92 from $15.
[tex]\frac{10.08}{0.21} =\frac{0.21x}{0.21}[/tex] - We divide by $0.21 to solve for x.
[tex]48=x[/tex]
And here, we get the exact same answer, 48 yards of ribbon.
A boy goes to school by first taking a bus for 1 3/4 km and then by walking 1/3 km. Find the distance of his house from the school.
The boy goes to school by bus for 1 3/4km, then he walks 1/3 km.
To determine the total distance he traveled you have to add both distances:
[tex]1\frac{3}{4}+\frac{1}{3}[/tex]To solve this sum, add the fractions first and then add the result to the whole number:
- Add both fractions:
[tex]\frac{3}{4}+\frac{1}{3}[/tex]To add both fractions you have to express them using the same denominator first. A common multiple between the denominators "4" and "3" is "12". Multiply the first fraction by 3 and the second by 4 to express them as their equivalent fractions with denominator 12. Then proceed to add them:
[tex]\frac{3\cdot3}{4\cdot3}+\frac{1\cdot4}{3\cdot4}=\frac{9}{12}+\frac{4}{12}=\frac{9+4}{12}=\frac{13}{12}[/tex]The result is 13/12, as you can see the numerator is greater than the denominator, which indicates that this is an improper fraction, i.e. its value is greater than 1. You can write this fraction as a mixed number as follows:
- Solve the division:
[tex]13\div12=1.08\bar{3}[/tex]The mixed number will have the whole number "1".
- To express the decimal value as a fraction, multiply it by 12
[tex]0.08\bar{3}\cdot12=1[/tex]The result is the numerator of the fraction, and the denominator will be 12, so:
[tex]0.08\bar{3}=\frac{1}{12}[/tex]And the resulting mixed number is:
[tex]\frac{13}{12}=1\frac{1}{12}[/tex]Finally, add the remaining whole number from the first sum to determine the distance between his house and the school:
[tex]1+1\frac{1}{12}=2\frac{1}{12}[/tex]The distance he traveled from home to school is 2 1/12 km.
Im confused on how to make the table and plug the dots while also describing both behaviors on this equation.
Here, we want to complete the table
To do this, we consider points on the plot
From what we have;
We are told that a graph of an exponential function does not cross the x-axis and thus, y cannot be zero
When x =0, y = 1
When x = 1, y = 2
when x = 2, y = 4
The y-intercept is the value of y when x = 0; it is the point at which the graph crosses the y-axis
What we have here is that wehn x = 0, y = 1
Hence, 1 is the y-intercept
Now, let us take a look at the end behavior
We can obtain this from the graph;
As x moves towards infinity, the y value moves towards infinity too as evident from the upward curve of the graph
As x moves toward negative infinity, y moves closer to zero
4x- 1/2+ 2/3x combine like terms
Okay, here we heve this:
We need to combine like terms in the following equation:
[tex]\begin{gathered} 4x-\frac{1}{2}+\frac{2}{3}x \\ =(4x+\frac{2}{3}x)-\frac{1}{2} \\ =\frac{14}{3}x-\frac{1}{2} \end{gathered}[/tex]how much ice pop mixture can each mold hold when full?
Explanation:
To know how much ice pop mixture can each mold hold, we need to calculate the volume of the mold.
The volume of a cone is equal to
[tex]V=\frac{1}{3}\pi r^2h[/tex]Where r is the radius and h is the height of the cone. Replacing r = 2 cm and h = 15 cm, we get:
[tex]\begin{gathered} V=\frac{1}{3}\pi(2cm)^2(15cm) \\ V=\frac{1}{3}\pi(4cm^2)(15cm) \\ V=20\pi cm^3 \end{gathered}[/tex]Therefore, the answer is
A. 20
What is the explicit formula for the sequence?3,1,-1, -3, -5,...a,= -2n +5a, = 17-5an= 2n-5an= -2n + 3
We need the formula that gives us the values of the sequence
where n is the value of the position
using the formula
[tex]a_n=-2n+5[/tex][tex]\begin{gathered} a_1=-2(1)+5=3 \\ a_2=-2(2)+5=1 \\ a_3=-2(3)+5=-1 \\ a_4=-2(4)+5=-3 \\ a_5=-2(5)+5=-5 \end{gathered}[/tex]as we can see the formula is the correct formula because we obtain the values of the sequence
Use the tangent to find the length of side PR. Express your answer to the nearest tenth. P 559 The length of side PR is approximately units.
tan (Q) = opposite/ adjacent
tan (55º) = PR/ 4.9
________________________
1.43 = PR/ 4.9
PR= 1.4* 4.9 = 6.9
Answer
6.9
______________________________________
Can you see the updates?
Do you have any questions regarding the solution?
____________________
PR= tan (55)* 4.9 = 6.997925 ≅ 7
_________________________________
what is x^4 − 14x2 + 45 as factored
Answer: B=3
Step-by-step explanation:
Given the set of all even integers between and including -18 to -6 , what is the probability of choosing a multiple of -6 from this set?1/93/74/9
TheseGiven the set: {-18, -16, -14, -12, -10, -8, -6}
This is 7 numbers
Multiplies of -6 are:
[tex]\begin{gathered} -6\times1=-6 \\ -6\times2=-12 \\ -6\times3=-18 \end{gathered}[/tex]This is 3 numbers
Therefore, the probability is given by:
[tex]P=\frac{multiplies\text{ of }-6}{total\text{ set }}[/tex]So:
[tex]P=\frac{3}{7}[/tex]Answer: 3/7
-Fractions-My sister needs help with this, and I totally forgot how to do fractions Mind helping out?
Because we have the same denominator we can do the subtraction
[tex]\frac{12}{10}-\frac{3}{10}=\frac{12-3}{10}=\frac{9}{10}[/tex]Paul did well the representation of the fractions in the diagram, but the operation that he made as we can see is wrong because the result is 9/10
Graph the line... running through: (1,3) with m=3
First, you have to locate the point (1,3)
Next, from that point, you have to locate the next one. To do that, you need the slope, which in this case is 3. So, from (1,3) you have to move 1 unit to the right and 3 units up, reaching the point (2, 6). Finally, you draw the line that passes through these two points
Rebecca makes four payments a year of $255 each for life
insurance; two payments of $455.35 each for real estate taxes;
and six payments of $66.21 each for auto insurance. How
much must Rebecca put into fixed savings each month to
cover her annual expenses for life insurance, auto insurance
and real estate taxes?
The amount Rebecca has put into fixed savings each month to cover her annual expenses for life insurance, auto insurance and real estate taxes is $ 194.
Given that:-
Amount invested in life insurance = $ 255
Number of payments in life insurance = 4
Amount invested in real estate taxes = $ 455.35
Number of payments in real estate taxes = 2
Amount invested in auto insurance = $ 66.21
Number of payments in auto insurance = 6
We have to find the amount Rebecca has put into fixed savings each month to cover her annual expenses for life insurance, auto insurance and real estate taxes.
Hence,
Total amount put by Rebecca in a year = 255*4 + 455.35*2 + 66.21*6 = 1020 + 910.70 + 397.26 = $ 2,327.96
Amount put by Rebecca in a month = 2327.96/12 = $ 193.997 ≈ $ 194
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This is a graph of the motion of a small boat traveling at a constant speed. Total Distance Traveled 12 10 8 Distance (Kilometers) 6 0 2 4 6 8 1012 Time (Hours) How far will the boat travel in 15 hours? O A. 25 km O B. 30 km O C. 15 km O D. 10 km
The speed of the boat is given by the gradient of the line.
Given two points (x1,y1) and (x2,y2) on a graph, the gradient of the line that passes through the two points is given by
[tex]\text{ gradient = }\frac{y_2-y_1}{x_2-x_1}[/tex]In this case,
the line passes through the points (0,0) and (2,2).
We can set (x1,x2) = (0,0) and (y1,y2) = (2,2)
Therefore,
[tex]\begin{gathered} \text{ gradient = }\frac{2-0}{2-0}=\frac{2}{2}=1 \\ \text{therefore} \\ \text{the speed = 1km/h} \end{gathered}[/tex]Given that a body travels with speed, s, and in time, t, the distance travelled, d, is given by
[tex]d=st[/tex]In this case,
s = 1km/h, and t = 15hours
Therefore,
[tex]d=1\times15=15\operatorname{km}[/tex]Therefore, the boat travelled a distance of 15km
Lauren was going to by her mom her favorite perfume for Christmas at a price of $31.95. She waited until it got too close to Christmas and the price went up to $41.49. What was the percent of increase in the price?
The percent of increase in the cost of the perfume is 29.86%
What is percentage and how can it be calculated?
A percentage is a figure or ratio that can be stated as a fraction of 100 in mathematics. If we need to determine a percentage of a number, multiply it by 100 and divide it by the total. So, a part per hundred is what the percentage refers to. Percent signifies for every 100. The sign "%" is used to denote it. No dimension exists for percentages. Thus, it is referred to as a dimensionless number.
Mathematically,
Percent of increase = [(Final value - Initial value)/(Initial value)]×100
Given, the final value of the perfume at purchase = $41.49
Also, the initial value of the perfume as assessed = $31.95
Therefore using the formula established in the literature above,
Percentage increase = [(41.49 - 31.95)/31.95]×100 = 29.86%
Thus, the percent of increase in the cost of the perfume is 29.86%
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I need help on writing the table and graphing it please !!
Given:
[tex]f(x)=2-\sqrt[]{x+6}[/tex]Calculate the values for f(x),
[tex]\begin{gathered} \text{for x=-6 , f(-6)=}2-\sqrt[]{-6+6} \\ f(-6)=2 \\ \text{for x=3 ,f(6)=2-}\sqrt[]{3+6} \\ f(3)=2-3=-1 \\ \text{for x=-2 f(-2)=2-}\sqrt[]{-2+6} \\ f(-2)=2-2=0 \\ \text{for x=1, f(1)=2-}\sqrt[]{1+6} \\ f(1)=-0.6 \end{gathered}[/tex]The graph of given function is,
7. Martha baked 332 muffins. She packed them in boxes of twelve muffins and sold each full box for S6How much money did she make?
• Given that Martha baked 332 muffins,
,• 332 / 12 = 27.67 boxes
,• If she sold each box for $ 6 ; it will be
27*6 = $162
0.67 * 6 = 4.02
Total 162+4.02 = 166.02≈ $166
• She will make $166, ,
Find the average rate of change of the following equation on the given interval. y=3x+1 on [45,48]
Given:
The equation is given as,
[tex]y=3x+1[/tex]The interval is given as,
[tex]\lbrack a,b\rbrack=\lbrack45,48\rbrack[/tex]The objective is to find the average rate of change.
Explanation:
The general formula to find the average rate of change is,
[tex]A=\frac{f(b)-f(a)}{b-a}\text{ . . . . (1)}[/tex]On plugging the function in the equation (1),
[tex]A=\frac{(3(48)+1)-(3(45)+1)}{48-45}[/tex]On further solving the above equation,
[tex]\begin{gathered} A=\frac{145-136}{3} \\ =\frac{9}{3} \\ =3 \end{gathered}[/tex]Hence, the average rate of change is 3.
Chris took four math quizzes and achieved a 68, 90, 95, and 75. What is his mean quiz average?
The average of a set is computed as follows:
[tex]\text{Average = }\frac{Tota\text{l sum of all numbers}}{\text{ number of items in the set}}[/tex]In this case,
[tex]\text{Average =}\frac{68+90+95+75}{4}=\frac{328}{4}=82[/tex]Find the area round to two decimal places as needed
To find the area of an obtuse triangle you have to multiply the base of the triangle by the vertical height and divide the result by 2 following the formula:
[tex]A=\frac{b\cdot h}{2}[/tex]The base of the triangle is b= 7 miles and the height is h= 8 miles, using these lengths calculate the area as follows:
[tex]\begin{gathered} A=\frac{7\cdot8}{2} \\ A=\frac{56}{2} \\ A=28mi^2 \end{gathered}[/tex]The area of the triangle is 28 square miles.
Find a degree 3 polynomial with real coefficients having zeros 3 and 1 - 32 and a lead coefficient of 1.
Write Pin expanded form. Be sure to write the full equation, including P(x)
The polynomial function of least degree with only real coefficients will be; y = x³ - 8 · x² + 22 · x - 20.
What is polynomial ?Algebraic expressions called polynomials include constants and indeterminates. Polynomials can be thought of as a type of mathematics.
The statement indicates that the polynomial has real coefficients having zeros 3 and 1 - 32 and a lead coefficient of 1.
By algebra of quadratic equations, equations with real coefficients with complex roots are α + i β and α - i β. Then we get;
y = 1 · (x - 3) · (x - 3 - i) · (x - 3 + i)
y = (x - 3) · [x² - 3 · x - i · x - 3 · x + i · x + (3 + i) · (3 - i)]
y = (x - 3) · (x² - 6 · x + 9 - i²)
y = (x - 3) · (x² - 6 · x + 10)
y = x³ - 6 · x² + 10 · x - 2 · x² + 12 · x - 20
y = x³ - 8 · x² + 22 · x - 20
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Answer:
[tex]p(x)=x^3-5x^2+16x-30[/tex]
Step-by-step explanation:
Given information:
Degree 3 polynomial with real coefficients.Zeros: 3 and (1 - 3i).Lead coefficient of 1.For any complex number [tex]z = a+bi[/tex] , the complex conjugate of the number is defined as [tex]z^*=a-bi[/tex].
If f(z) is a polynomial with real coefficients, and z₁ is a root of f(z)=0, then its complex conjugate z₁* is also a root of f(z)=0.
Therefore, if p(x) is a polynomial with real coefficients, and (1 - 3i) is a root of p(x)=0, then its complex conjugate (1 + 3i) is also a root of p(x)=0.
Therefore, the polynomial in factored form is:
[tex]p(x)=a(x-3)(x-(1-3i))(x-(1+3i))[/tex]
As the leading coefficient is 1, then a = 1:
[tex]p(x)=(x-3)(x-(1-3i))(x-(1+3i))[/tex]
Expand the polynomial:
[tex]\implies p(x)=(x-3)(x-(1-3i))(x-(1+3i))[/tex]
[tex]\implies p(x)=(x-3)(x-1+3i)(x-1-3i)[/tex]
[tex]\implies p(x)=(x-3)(x^2-x-3xi-x+1+3i+3ix-3i-9i^2)[/tex]
[tex]\implies p(x)=(x-3)(x^2-x-x-3xi+3ix+1+3i-3i-9i^2)[/tex]
[tex]\implies p(x)=(x-3)(x^2-2x+1-9(-1))[/tex]
[tex]\implies p(x)=(x-3)(x^2-2x+10)[/tex]
[tex]\implies p(x)=x^3-2x^2+10x-3x^2+6x-30[/tex]
[tex]\implies p(x)=x^3-2x^2-3x^2+10x+6x-30[/tex]
[tex]\implies p(x)=x^3-5x^2+16x-30[/tex]
In circle F with mZEFG = 30 and EF = 4 units, find the length of arc EG.. 4Round to the nearest hundredth.
The arc length can be found through the formula:
[tex]s=2\ast\pi\ast r\ast\frac{\theta}{360}[/tex]then, we can say that r is equal to 4 and the angle is 30°
[tex]\begin{gathered} s=2\ast\pi\ast4\ast\frac{30}{360} \\ s\approx2.09 \end{gathered}[/tex]Answer:
The arc length is approximately equal to 2.09
Describe and justify the methods you used to solve the quadratic equations in parts A and B.
We know that give any pair of real numbers A and B, the following statement will be true:
[tex]A\cdot B=0[/tex]if, and only if
[tex]\begin{gathered} A=0 \\ or \\ B=0 \end{gathered}[/tex]Now, if we factor a quadratic equation into two factors A and B, and use the fact we've just mentioned, we can then equal each factor to zero, solve for x and get the solutions to said quadratic equation.
g(n) = n2 − 4
h(n) = n − 5
Find g(n) · h(n)
g(x) = 4x + 4
f(x) = x3 − 1
Find (g ◦ f)(x)
The value of
g(n) · h(n) = n³ - 5n² - 4n + 20 (g ◦ f)(x) = 4x³What is function?The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.
Given:
g(n) = n² − 4, h(n) = n − 5
g(n).h(n)
= (n² − 4).(n-5)
= n³ - 5n² - 4n + 20
and, g(x) = 4x + 4, f(x) = x³ − 1
(gof)(x)
=g(f(x))
=g(x³-1)
= 4(x³-1) + 4
= 4x³ - 4 + 4
= 4x³
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A baker need 2/3 cup of sugar,but he can only find a 1/2 cup measure,so he decides to estimate, Which of the following would result in the correct amount of sugar?A)One Full scoop plus 1/3 of a scoopB)One Full scoop plus 1/2 of a scoop C) Two ScoopsD)3/4 of a scoop
He needs 2/3 cup of sugar . But he can only find 1/2 cup measures.
I'll send in pictures of the question questions 2 goes with number 1
Since the equation is y=3/8x and x is equal to 44/3, we have
[tex]\begin{gathered} y=\frac{3}{8}\cdot\frac{44}{3}=\frac{132}{24} \\ \frac{132}{24}=\frac{66}{12}=\frac{33}{6}\text{ Simplifying} \\ \frac{33}{6}=5.5\text{ Dividing} \\ \text{Answer is: }y=5.5 \end{gathered}[/tex]