Let 'x' be the number of calories per cup of popcorn, and 'y' be the number of calories per ounce of soda.
Given that 3 cups of popcorn and 6 oz of soda constitute 246 calories,
[tex]3x+6y=246[/tex]Also given that 1 cups of popcorn and 14 oz of soda constitute 274 calories,
[tex]x+14y=274[/tex]Solve the equations using Elimination Method.
Subtract 3 times equation 2 from equation 1,
[tex]\begin{gathered} (3x+6y)-3(x+14y)=246-3(274) \\ 3x+6y-3x-42y=246-822 \\ -36y=-576 \\ y=16 \end{gathered}[/tex]Substitute this value in equation 1, to obtain 'x' as,
[tex]\begin{gathered} 3x+6(16)=246 \\ 3x+96=246 \\ 3x=150 \\ x=50 \end{gathered}[/tex]Thus, the solution of the system of equations is x=50 and y=16.
Therefore, there are 50 calories per cup of popcorn, and 16 calorie per ounce of soda.
How can you represent Pattered from every day life by using tables,expressions and graphs
For example, we can look at the variations of temperature by the time of the day.
We can write it in a two column table, where we can write the hour in one column and the temperature in the other column.
This will show us a relationship between them that is oscillating.
We can graph this and have something like:
Then, we can adjust a function to that, like a trigonometrical function that can model this relation between temperature and hour of the day. There you wil have an expression for this pattern.
Need help confirming my answer, do I just put x=1 or x=1,-3/2
Applying quadratic formula to the given quadratic equation, we get the solutions as [tex]x=1,-\frac{3}{2}[/tex].
It is given to us that the quadratic equation is -
[tex]-2x^{2} -1x+3=0[/tex] ---- (1)
We have to solve by this by quadratic formula.
From equation (1), we have
[tex]-2x^{2} -1x+3=0\\= > -2x^{2} -x+3=0\\= > -(2x^{2} +x-3)=0\\= > 2x^{2} +x-3=0[/tex]----- (2)
The above equation (2) is in the form of a quadratic equation
[tex]ax^{2} +bx+c=0[/tex]
where, a = 2
b = 1
and, c = -3
Now, using the quadratic formula, we know
[tex]x=\frac{-b+\sqrt{b^{2} -4ac} }{2a}[/tex] and, [tex]x=\frac{-b-\sqrt{b^{2} -4ac} }{2a}[/tex]
Substituting the values of a, b, and c in the above formulas to find the value of x, we get
[tex]x=\frac{-b+\sqrt{b^{2} -4ac} }{2a} and x=\frac{-b-\sqrt{b^{2} -4ac} }{2a}\\= > x=\frac{-1+\sqrt{(-1)^{2} -4*2*(-3)} }{2*2} and x=\frac{-1-\sqrt{(-1)^{2} -4*2*3} }{2*2}\\= > x=\frac{-1+\sqrt{25} }{4} and x=\frac{-1-\sqrt{25} }{4}\\= > x= \frac{-1+5}{4} and x=\frac{-1-5}{4} \\= > x=\frac{4}{4} and x=\frac{-6}{4} \\= > x=1 and x= -\frac{3}{2}[/tex]
Thus, solving the given quadratic equation through quadratic formula, we get the solutions as [tex]x=1,-\frac{3}{2}[/tex].
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The function f(x)=2,500(1.012)^x represents the amount, in dollars, in a savings account after x years. Which statement is true? A. The account earns 0.12% interest per year. B. The account earns 0.012% interest per year. C. The initial amount in the account was $2.500. D. The amount in the account increases by $2,500 each year,
Answer:
C. The initial amount in the account was $2,500.
Explanation:
The function that represents the amount, in dollars, in a savings account after x years is given as:
[tex]f\mleft(x\mright)=2,500\mleft(1.012\mright)^x[/tex]When x=0 (Initially)
[tex]\begin{gathered} f\mleft(0\mright)=2,500\mleft(1.012\mright)^0 \\ =2,500\times1 \\ =\$2,500 \end{gathered}[/tex]Therefore, the initial amount in the account was $2,500.
8. Which of the following ordered pairs is a solution to f(x) = 1/2x -8?(4, - 4)(2, - 7)(10, 3)(-6, 11)
ANSWER
(2, -7)
EXPLANATION
We want to find which of the ordered pairs is a solution for:
[tex]f(x)\text{ = }\frac{1}{2}x\text{ - 8}[/tex]Ordered pairs are usually given in the form (x, f(x)). That is the value of x and the value of the function of x.
We have to put each of the first values in the ordered pairs in the given function and see if it results in the second value.
=> (4, -4)
[tex]\begin{gathered} f(4)\text{ = }\frac{1}{2}(4)\text{ - 8} \\ f(4)\text{ = 2 }-8 \\ f(4)\text{ = -}6 \end{gathered}[/tex]Not a solution
=> (2, -7)
[tex]\begin{gathered} f(2)\text{ = }\frac{1}{2}(2)\text{ - 8} \\ f(2)\text{ = }1\text{ - 8} \\ f(2)\text{ = -}7 \end{gathered}[/tex]This is a solution.
=> (10, 3)
[tex]\begin{gathered} f(10)\text{ = }\frac{1}{2}(10)\text{ - 8} \\ f(10)\text{ = }5\text{ - 8} \\ f(10)\text{ = }-3 \end{gathered}[/tex]Not a solution.
=> (-6, 11)
[tex]\begin{gathered} f(-6)\text{ = }\frac{1}{2}(-6)\text{ - 8} \\ f(-6)\text{ = -3 - 8} \\ f(-6)\text{ = -11} \end{gathered}[/tex]Not a solution.
The only solution there is (2, -7)
I need help with getting to the answer to number 6
We have the following pair of functions:
[tex]\begin{gathered} f(x)=x^3+6x \\ g(x)=\sqrt{8x} \end{gathered}[/tex]And we need to find (fog)(2). In order to do this we can start by calculating the composite function (fog)(x)=f(g(x)). Its expression is given by taking the equation of f(x) and replacing x with the expression of g(x). Then we get:
[tex]\begin{gathered} (f\circ g)(x)=f(g(x))=g(x)^3+6g(x)=(\sqrt{8x})^3+6\sqrt{8x} \\ (f\circ g)(x)=(\sqrt{8x})^3+6\sqrt{8x} \end{gathered}[/tex]We need to find (fog)(2) so we just need to take x=2 in the equation above:
[tex]\begin{gathered} (f\circ g)(2)=(\sqrt{8\cdot2})^3+6\sqrt{8\cdot2} \\ (f\circ g)(2)=(\sqrt{16})^3+6\cdot\sqrt{16} \\ (f\circ g)(2)=4^3+6\cdot4 \\ (f\circ g)(2)=64+24 \\ (f\circ g)(2)=88 \end{gathered}[/tex]AnswerThen the answer is 88.
The fox population in a certain region has a continuous growth rate of 9 percent per year.
SOLUTION
The function can be derived from the model
[tex]\begin{gathered} P=P_oe^{(\ln r)t^{}} \\ \\ r\text{ here represents 1 + 9 percent growth rate } \end{gathered}[/tex]So the function becomes
[tex]P(t)=2000_{}e^{(\ln 1.09)t}[/tex]So the fox population in 2008
2008 - 2000 = 8
So our t becomes 8
The population becomes
[tex]\begin{gathered} P=2000_{}e^{(\ln 1.09)t} \\ P=2000_{}e^{(\ln 1.09)\times8} \\ P=\text{ }2000_{}e^{0.086177\times8} \\ =2000_{}e^{0.6894} \\ =\text{ 3985.04} \end{gathered}[/tex]So the Population = 3985
Is this a function or non-function {(3,4),(4,-6),(5,-7),(3,2),(-2,5)}
Recall that a set of ordered pairs A represents a function if:
[tex](x,y),(x,z)\in A\text{ if and only if y=z.}[/tex]Now, notice that (3,4) and (3,2) are in the given set of ordered pairs, since
[tex]4\ne2[/tex]we get that the given set does not correspond to a function.
Answer: Non-function.
is 13/4 and 21/4 equivalent an equivalent fraction?
To determine if they are equivalent we equal them and if when simplified they do not show the same values, they are not. That is:
[tex]\frac{13}{4}=\frac{21}{4}\Rightarrow3.25\ne5.25[/tex]From that, we can see that they are not equivalent fractions.
A litter of kittens consists of one gray female, two gray males, two black females and one black male. You randomly pick one kitten, what is the probability it is black?
Total number of kittens = 6
Gray kittens= 1 female+2 males = 3
Black kittens= 2 female+ 1 male =3
Probability of picking one black kitten = black kittens/ total kittens = 3/6 =1/2
Albert has 16 oz of cheddar cheese and 8 oz of mozzarella cheese. He used 5 1/2 oz of the cheddar and 3 1/3 oz of the mozzarella cheese in a recipe. What is the total amount of cheese that Albert has left?
PROBLEM
Total
16 cheddar cheese
8 mozzarella cheese
Solution
He uses
[tex]\begin{gathered} 5\frac{1}{2}\text{ of cheddar out of 16} \\ 3\frac{1}{3}\text{ of mozzarella out of 8} \end{gathered}[/tex][tex]\begin{gathered} \\ \text{Cheddar left = 16 - 5}\frac{1}{2}\text{ = 10}\frac{1}{2} \end{gathered}[/tex][tex]\text{Mozzarella left = 8 - 3}\frac{1}{3}\text{ = 4}\frac{2}{3}[/tex][tex]\begin{gathered} \\ \text{Total ch}eese\text{ left } \\ = \end{gathered}[/tex][tex]\begin{gathered} \\ =\text{ 10}\frac{1}{2}\text{ - 4}\frac{2}{3} \end{gathered}[/tex][tex]=\text{ 5}\frac{5}{6}[/tex]751 body temperature measurements were taken. The sample data resulted in a sample mean of 98.1 F and a sample standard deviation of 0.7 F. Use the traditional method and a 0.05 significance level to test the claim that the mean body temperature is less than 98.6 F.
The mean value of the sample is 98.1 F and its standard deviation is 0.7 F.
The margin of error of the mean value is given by 0.7/sqrt(751) = 0.7/27.4 = 0.026 (rounded to the nearest thousandth)
Using the Z test, we got: Z = (98.6 - 98.1)/(0.026) = 0.5/0.026 = 196
Therefore, the mean value of the sample is incompatible with 98.6 and we can claim that the mean body temperature is less than it.
Which equation is correct? (6 points)Group of answer choicessec x° = opposite ÷ adjacentcot x° = opposite ÷ adjacentcosec x° = opposite ÷ adjacentsec x° = hypotenuse ÷ adjacent
Answer:
Concept:
To figure this question out, we will use the trigonometric ratios below
SOH CAH TOA
[tex]\begin{gathered} SOH \\ sin\theta=\frac{opposite}{hypotenus}=S=\frac{O}{H} \\ \cos\theta=\frac{adjacent}{hypotenus}=C=\frac{A}{H} \\ \tan\theta=\frac{opposite}{adjacent}=T=\frac{O}{A} \end{gathered}[/tex]Using the inverse trigonometric identity,
[tex]\begin{gathered} cosecx^0=\frac{1}{sinx^0} \\ secx^0=\frac{1}{cosx^0} \\ cotx^0=\frac{1}{tanx^0} \end{gathered}[/tex]By simplifying further, we will have that
[tex]\begin{gathered} cosecx^0=\frac{1}{s\imaginaryI nx^{0}} \\ cosecx^0=\frac{1}{\frac{opposite}{hypotenu}}=1\times\frac{hypotenus}{opposite} \\ cosecx^0=\frac{hypotenus}{opposite} \end{gathered}[/tex][tex]\begin{gathered} secx^{0}=\frac{1}{cosx^{0}} \\ secx^0=\frac{1}{\frac{adjacent}{hypotenus}}=1\times\frac{hypotenus}{adjacent} \\ secx^0=\frac{hypotenus}{adjacent} \end{gathered}[/tex][tex]\begin{gathered} cotx^{0}=\frac{1}{tanx^{0}} \\ cotx^0=\frac{1}{\frac{opposite}{adjacent}}=1\times\frac{adjacent}{opposite} \\ cotx^0=\frac{adjacent}{oppos\imaginaryI te} \end{gathered}[/tex]Hence,
The final answer is
[tex]\Rightarrow secx^0=\frac{hypotenus}{adjacent}[/tex]Write an equation for the line parallel to the given line that contains B. B(3, 8) ; y = - 4x + 7
You have to write an equation parallel to the line: y=-4x+7 that crosses the point (3,8)
One characteristic that two parallel lines share is that they have the same slope.
The slope for the known line corresponds to the coefficient multiplying the x-term and is m=-4
The line you have to find must have the same slope.
Using the point-slope form you can determine said line. The general structure is:
[tex]y-y_1=m(x-x_1)[/tex]Where
m is the slope
x₁, y₁ are the coordinates of a point crossed by the line.
Using (3, 8) and m=-4
[tex]y-8=-4(x-3)[/tex]Now you have to solve it and write it in slope-intercept form. First step is to solve the term in parentheses by applying the distributive propperties of multiplication:
[tex]\begin{gathered} y-8=-4x-3\cdot(-4) \\ y-8=-4x+12 \end{gathered}[/tex]Next pass "-8" to the other side of the equal sign:
[tex]\begin{gathered} y-8+8=-4x+12+8 \\ y=-4x+20 \end{gathered}[/tex]The equation for the line parallel to y=-4+7 is y=-4x+20
NEED HELP DUE BY WEDNESDAY OR TOMMOROW. Solve each of the equations and select the numbers that represent solutions to more than one of the six equations. Select all that apply. 4x-3=17 8(x + 1) = 24 5(x - 2) = 20 34 - 7x = 20 31 - x = 29 3x +6=21. A. x=1. B. x=2. C. x=3. D. x=4.E. x=5. F. X = 6.
Given
The equations,
4x-3=17, 8(x + 1) = 24, 5(x - 2) = 20, 34 - 7x = 20, 31 - x = 29, 3x +6=21.
To find the solution to each equations.
Explanation:
It is given that,
The equations,
4x-3=17, 8(x + 1) = 24, 5(x - 2) = 20, 34 - 7x = 20, 31 - x = 29, 3x +6=21.
That implies,
1)
[tex]\begin{gathered} 4x-3=17 \\ 4x=17+3 \\ 4x=20 \\ x=\frac{20}{4} \\ x=5 \end{gathered}[/tex]Hence, the solution is x=5.
2)
[tex]\begin{gathered} 8(x+1)=24 \\ x+1=\frac{24}{8} \\ x+1=3 \\ x=3-1 \\ x=2 \end{gathered}[/tex]Hence, the solution is x=2.
3)
[tex]\begin{gathered} 5(x-2)=20 \\ x-2=\frac{20}{5} \\ x-2=4 \\ x=4+2 \\ x=6 \end{gathered}[/tex]Hence, the solution is x=6.
4)
[tex]\begin{gathered} 34-7x=20 \\ 34-20=7x \\ 7x=14 \\ x=\frac{14}{7} \\ x=2 \end{gathered}[/tex]Hence, the solution is x=2.
5)
[tex]\begin{gathered} 31-x=29 \\ 31-29=x \\ x=2 \end{gathered}[/tex]Hence, the solution is x=2.
6)
[tex]\begin{gathered} 3x+6=21 \\ 3x=21-6 \\ 3x=15 \\ x=\frac{15}{3} \\ x=5 \end{gathered}[/tex]Hence, the solution is x=5.
Which is the measure of an interior angle of a regular decagon?30°36°144°150°
SOLUTION:
We are to find the measure of an interior angle of a regular decagon.
A decagon is a plane figure with ten straight sides and angles.
To find the sum of interior angles in a decagon;
(n - 2) x 180 (where n = 10)
(10 - 2) x 180
= 8 x 180
= 1440 degrees
The measure of an interior angle of a regular decagon is;
1440 / 10
144 degrees
How much interest in dollars is earned in 5 years on $8,200 deposited in an account paying 6% interest compounded semiannually round to the nearest cent
Using compound interest formula:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Where:
A = Amount
P = Principal = 8200
r = Interest rate = 6% = 0.06
n = Number of times interest is compounded per year = 2
t = time = 5
so:
[tex]\begin{gathered} A=8200(1+\frac{0.06}{2})^{2\cdot5} \\ A=11020.11 \end{gathered}[/tex]Therefore, the interest is the amount minus the amount invested:
[tex]\begin{gathered} I=A-P \\ I=11020.11-8200 \\ I=2820.11 \end{gathered}[/tex]Answer:
$2820.11
Felipe is a software salesman. His base salary is $2100, and he makes an additional $70 for every copy of math is fun he sells. Let P represent his total pay in dollars, and let N represent the number of copies of math is fun he sells. Write an equation relating P to N then use this equation to find his total pay if he sells 27 copies of math is fun.
Based on the given information, we can determine the constant and the variable payment.
• He gets $2100 as based salary (never changes, constant).
,• The additional $70 depends on ,N ,(variable).
This information can help us build the equation:
[tex]P=70N+2100[/tex]Therefore, if he sells 27 copies, then the total payment is:
[tex]P=70\cdot27+2100[/tex][tex]P=1890+2100[/tex][tex]P=3990[/tex]Answer:
[tex]P=70N+2100[/tex]P( 27 ) = $3990
Larry Mitchell invested part of his $22,000 advance at 2% annual simple interest and the rest at 6% annual simple interest if his total yearly interest from both accounts was $760 find the amount invested at each rate The amount invested at 2%The amount invested at 6%( please don’t need an detailed explanation just the answer)
Larry Mitchell invested part of his $22,000 advance at 2% annual simple interest and the rest at 6% annual simple interest if his total yearly interest from both accounts was $760 find the amount invested at each rate
The amount invested at 2%
The amount invested at 6%
Let
x -----> amount invested at 2%
(22,000-x) -----> amount invested at 6%
we have that
x(0.02)+(22,000-x)(0.06)=760
solve for x
0.02x+1,320-0.06x=760
0.06x-0.02x=1,320-760
0.04x=560
x=14,000
(22,000-14,000)=8,000
therefore
amount invested at 2% -----> $14,000amount invested at 6% -----> $8,000The sum of two numbers is 60. The greater number is 6 more than the smaller number which equation can be used to solve for the smaller number
x ----> is the smaller number
x+6 ----> is the greater number
the equation is
[tex]x+(x+6)=60[/tex]Find the mean with and without the outlier: 66, 55, 65, 44, 54, 10
Answer:
To calculate the mean of the set of values with the outlier, we will use the formula below
Concept:
An outlier is an observation that lies an abnormal distance from other values in a random sample of a population.
The mean with the outlier will be
[tex]mean=\frac{total\text{ addition of numbers}}{number\text{ of data}}[/tex]By substituting the value, we will have
[tex]\begin{gathered} mean=\frac{66+55+65+44+54+10}{6} \\ mean=\frac{294}{6} \\ mean=49 \end{gathered}[/tex]Hence,
The mean of the data with the outlier is = 49
the table shows the probability distrubution of a random variable Z.Z- -17, -16,-15,-14,-13 P(Z)- 0.02 , 0.73, 0.02, 0.08, 0.15what is the mean of the probability distrubution
In order to find the mean, we have to multiply each value with its probability and then add al the results:
Step 1- multiplying each value by its probabilityStep 2 - adding all the resultsNow, we add all the results we found on the previous step:
Mean = -0.34 - 11.68 - 0.30 - 1.12 - 1.95
Mean = -15.39
Answer: mean = -15.39
Precalc and i need help withb. Sec(18pie)c. Sin(7pie/6) tan(8pie/3)d. Tan(pie/12)
In b we need to find:
[tex]\sec 18\pi[/tex]It's important to recal that the secant is equal to:
[tex]\sec 18\pi=\frac{1}{\cos18\pi}[/tex]Another important property that will be useful is:
[tex]\cos x=\cos (x+2\pi m)[/tex]Where m is any integer. Let's see if we can write 18*pi using this. We can take x=0 so we have:
[tex]\begin{gathered} 18\pi=x+2\pi m=2\pi m \\ 18\pi=2\pi m \end{gathered}[/tex]If we divide both sides by 2*pi:
[tex]\begin{gathered} \frac{18\pi}{2\pi}=\frac{2\pi m}{2\pi} \\ 9=m \end{gathered}[/tex]Since m is an integer then we can assure that:
[tex]\cos 18\pi=\cos (0+2\pi\cdot9)=\cos 0=1[/tex]Then the secant is given by:
[tex]\sec 18\pi=\frac{1}{\cos18\pi}=\frac{1}{\cos 0}=1[/tex]So the answer to b is 1.
In c we need to find:
[tex]\sin (\frac{7\pi}{6})\tan (\frac{8\pi}{3})[/tex]Here we can use the following properties in order to write those angles as angles of the first quadrant:
[tex]\begin{gathered} \sin (x)=-\sin (x-\pi) \\ \tan (x)=\tan (x-m\pi)\text{ with }m\text{ being an integer} \end{gathered}[/tex]So we have:
[tex]\begin{gathered} \sin (\frac{7\pi}{6})=-\sin (\frac{7\pi}{6}-\pi)=-\sin (\frac{\pi}{6}) \\ \tan (\frac{8\pi}{3})=\tan (\frac{8\pi}{3}-3\pi)=\tan (-\frac{1}{3}\pi) \end{gathered}[/tex]If we convert these two angles from radians to degrees by multiplying 360° and dividing by 2*pi we have:
[tex]\begin{gathered} \frac{\pi}{6}\cdot\frac{360^{\circ}}{2\pi}=30^{\circ} \\ -\frac{1}{3}\pi\cdot\frac{360^{\circ}}{2\pi}=-60^{\circ} \end{gathered}[/tex]And remeber that:
[tex]\tan x=-\tan (-x)[/tex]So we get:
[tex]\begin{gathered} \sin (\frac{7\pi}{6})=-\sin (\frac{\pi}{6})=-\sin (30^{\circ}) \\ \tan (\frac{8\pi}{3})=\tan (-\frac{\pi}{3})=-\tan (\frac{\pi}{3})=-\tan (60^{\circ}) \end{gathered}[/tex]Then we can use a table of values:
Then:
[tex]\sin (\frac{7\pi}{6})\tan (\frac{8\pi}{3})=\sin (30^{\circ})\cdot\tan (60^{\circ})=\frac{1}{2}\cdot\sqrt[]{3}=\frac{\sqrt[]{3}}{2}[/tex]So the answer to c is (√3)/2.
In d we need to find:
[tex]\tan (\frac{\pi}{12})[/tex]In order to do this using the table we can use the following:
[tex]\begin{gathered} \tan x=\frac{\sin x}{\cos x} \\ \sin 2x=2\sin x\cos x \\ \cos 2x=\cos ^2x-\sin ^2x \\ \cos ^2x+\sin ^2x=1 \end{gathered}[/tex]So from the first one we have:
[tex]\tan (\frac{\pi}{12})=\frac{\sin (\frac{\pi}{12})}{\cos (\frac{\pi}{12})}[/tex]We convert pi/12 into degrees:
[tex]\frac{\pi}{12}\cdot\frac{360^{\circ}}{2\pi}=15^{\circ}[/tex]So we need to find the sine and cosine of 15°. We use the second equation:
[tex]\begin{gathered} \sin 30^{\circ}=\frac{1}{2}=\sin (2\cdot15^{\circ})=2\sin 15^{\circ}\cos 15^{\circ} \\ \sin 15^{\circ}\cos 15^{\circ}=\frac{1}{4} \end{gathered}[/tex]Then we use the third:
[tex]\begin{gathered} \cos (30^{\circ})=\frac{\sqrt[]{3}}{2}=\cos (2\cdot15^{\circ})=\cos ^215^{\circ}-\sin ^215^{\circ} \\ \frac{\sqrt[]{3}}{2}=\cos ^215^{\circ}-\sin ^215^{\circ} \end{gathered}[/tex]And from the fourth equation we get:
[tex]\begin{gathered} \cos ^215^{\circ}+\sin ^215^{\circ}=1 \\ \sin ^215^{\circ}=1-\cos ^215^{\circ} \end{gathered}[/tex]We can use this in the previous equation:
[tex]\begin{gathered} \frac{\sqrt[]{3}}{2}=\cos ^215^{\circ}-\sin ^215^{\circ}=\cos ^215^{\circ}-(1-\cos ^215^{\circ}) \\ \frac{\sqrt[]{3}}{2}=2\cos ^215^{\circ}-1 \\ \cos 15^{\circ}=\sqrt{\frac{1+\frac{\sqrt[]{3}}{2}}{2}} \\ \cos 15^{\circ}=\sqrt{\frac{1}{2}+\frac{\sqrt[]{3}}{4}} \end{gathered}[/tex]So we found the cosine. For the sine we use the expression with the sine and cosine multiplying:
[tex]\begin{gathered} \sin 15^{\circ}\cos 15^{\circ}=\frac{1}{4} \\ \sin 15^{\circ}\cdot\sqrt[]{\frac{1}{2}+\frac{\sqrt[]{3}}{4}}=\frac{1}{4} \\ \sin 15^{\circ}=\frac{1}{4\cdot\sqrt[]{\frac{1}{2}+\frac{\sqrt[]{3}}{4}}} \end{gathered}[/tex]Then the tangent is:
[tex]\tan (15^{\circ})=\frac{\sin(15^{\circ})}{\cos(15^{\circ})}=\frac{1}{4\cdot\sqrt[]{\frac{1}{2}+\frac{\sqrt[]{3}}{4}}}\cdot\frac{1}{\sqrt[]{\frac{1}{2}+\frac{\sqrt[]{3}}{4}}}=\frac{1}{4}\cdot\frac{1}{\frac{1}{2}+\frac{\sqrt[]{3}}{4}}[/tex][tex]\tan (15^{\circ})=\frac{1}{4}\cdot\frac{1}{\frac{1}{2}+\frac{\sqrt[]{3}}{4}}=\frac{1}{2+\sqrt[]{3}}[/tex]Then the answer to d is:
[tex]\frac{1}{2+\sqrt[]{3}}[/tex]Use the number line diagram below to answer the following questions.1.What is the length of each segment on the number line?
Given from the number line that the total number of segments between 0 and 1 is 12 segments.
1) Therefore, the length of each segment on the number line is
[tex]\frac{1-0}{12}=\frac{1}{12}[/tex]Hence, the answer is
[tex]\frac{1}{12}[/tex]2) There are 8 segments between 0 and K.
Therefore, point K represents
[tex]\frac{1}{12}\times8=\frac{8}{12}=\frac{2}{3}[/tex]Hence, the answer is
[tex]\frac{2}{3}[/tex]3) The opposite of K is
[tex]-\frac{2}{3}\text{ since it falls on the negative side of the number line.}[/tex]Hence, the answer is
[tex]-\frac{2}{3}[/tex]Which sequence describes Ahmed's expected hourly wages, in dollars, starting with his current wage?
Since Ahmed will start with $7.50 per hour
Then the sequence must start with 7.50
Then the answer should be A or B or C
Since his hourly rate will increase by $0.25 per hour
Then the number in the sequence should be increased
Then the answer is B or C because A is decreasing
We have to add 0.25 to the first rate to get the second rate
[tex]\begin{gathered} 7.50+0.25=7.75 \\ 7.75+0.25=8.00 \\ 8.00+0.25=8.25 \\ 8.25+0.25=8.50 \end{gathered}[/tex]Then the correct answer is
$7.50, $7.75, $8.00, $8.25, $8.50
The answer is B
how do I solve a liner model ?
The values that are represented by the dots are close to the horizontal line, so they are a non-random pattern, because they follow the horizontal line without going to far from it
Since these points are all around the horizontal line, they also represent a linear model
So the answer for hte first box is "non-random" and the answer for the second box is "linear"
how many weeks does it take to empty the lake?
The rate of emptying the lake is -1/8.
The rate of filling the lake is 1/15
Let t be the time in weeks to empty the lake,
Now, add the given rate to get the total rate of emptying of -1/t.
[tex]\begin{gathered} \frac{-1}{8}+\frac{1}{15}=-\frac{1}{t} \\ \frac{-15+8}{120}=-\frac{1}{t} \\ -7\times t=-120 \\ t=\frac{120}{7} \end{gathered}[/tex]Thus,
[tex]t=17\frac{1}{7}[/tex]Therefore, it will take 17 weeks and 1 day to empty the lake.
fing the length of the missing side
The area is given as
[tex]x^2-6x+9[/tex]We can either divide the area by the side given and get the other side
OR
We can simply factorize the area and hence determine the two factors that were multiplied together. Note that one factor has already been given (that is x-3).
To factorize the polynomial;
[tex]\begin{gathered} x^2-6x+9 \\ =(x-3)(x-3) \end{gathered}[/tex]This means the other side is also (x - 3)
Find the measure of each labeled angle as well as the values of x, y, and z.
Notice that the angle labelled as 3y and the angle with a measure of 72° are supplementary angles. Then:
[tex]3y+72=180[/tex]Substract 72 from both sides of the equation:
[tex]\begin{gathered} 3y+72-72=180-72 \\ \Rightarrow3y=108 \end{gathered}[/tex]The angle labelled as x and the angle labelled as 3y are corresponding angles. Then, they have the same measure:
[tex]x=3y[/tex]Since 3y=108, then:
[tex]x=108[/tex]On the equation 3y=108, divide both sides by 3 to find the value of y:
[tex]\begin{gathered} \frac{3y}{3}=\frac{108}{3} \\ \Rightarrow y=36 \end{gathered}[/tex]Finally, notice that the angle labelled as 3z+18 and the angle labelled as x are corresponding angles. Then, they have the same measure:
[tex]3z+18=x[/tex]Substitute x=108 and isolate z to find its value:
[tex]\begin{gathered} \Rightarrow3z+18=108 \\ \Rightarrow3z=108-18 \\ \Rightarrow3z=90 \\ \Rightarrow z=\frac{90}{3} \\ \Rightarrow z=30 \end{gathered}[/tex]Therefore, the measure of the angles labelled as 3z+18, x and 3y is 108°. The values of x, y and z are:
[tex]\begin{gathered} x=108 \\ y=36 \\ z=30 \end{gathered}[/tex]108010 -8 -62IC-Find the slope of the line.Slope = m =Enter your answer as an integer or as a reduced fraction in the form A/B.Question Help: Video Message
The slope formula is givenb by:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]To get the slope from the graph, we will pick out two points lying on the line:
Point 1: (x, y) = (-6, 10)
Point 2: (x, y) = (0, -8)
We will then proceed to use these points to calculate the slope, we have:
[tex]\begin{gathered} m=\frac{-8-10}{0--6}=-\frac{18}{6} \\ m=-3 \end{gathered}[/tex]The slope (m) = -3
12. Write a paragraph proof.Given: AB = CD, BC = DAProve: AABC = ACDA
Answer:
Triangles ABC and CDA share the side AC, therefore they have three congruent sides. Since AB is congruent to CD and BC is congruent to DA then by the SSS criteria we get that triangles ABC and CDA are congruent.