The cashier has 7 bills of $10 and 17 bills of $20 (found using linear equation).
According to the question,
We have the following information:
A cashier has 24 bills, all of which are $10 or $20 bills. The total value of the money is $410.
Now, let's take the number of $10 bills to be x and the number of $20 bills to be y.
So, we have the following expression:
x+y = 24
x = 24-y .... (1)
10x+20y = 410
Taking 10 as a common factor from the terms on the left hand side:
10(x+2y) = 410
x+2y = 410/10
x+2y = 41
Now, putting the value of x from equation 1:
24-y+2y = 41
24+y = 41
y = 41-24
y = 17
Now, putting this value of y in equation 1:
x = 24-y
x = 24-17
x = 7
Hence, the cashier has 7 bills of $10 and 17 bills of $20 when the total value of the money is $410.
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You cut a piece of wood that is 69 inches long. The wood is cut into 3 pieces. The second piece is 8 inches
longer than the first. The third piece is 8 inches longer than the second piece. Find the length of each of
the three pieces.
The length of piece one will be 15 inch, the length of piece two will be 23 inch and the length of third piece will be 31 inch as per the given conditions of "You cut a piece of wood that is 69 inches long. The wood is cut into 3 pieces. The second piece is 8 inches longer than the first. The third piece is 8 inches longer than the second piece."
What is system of equation?A finite set of equations for which common solutions are sought is referred to in mathematics as a set of simultaneous equations, also known as a system of equations or an equation system.
What is equation?In mathematics, an equation is an expression or a statement that consists of two algebraic expressions that have the same value and are separated from one another by the equal symbol. It is an otherwise stated statement that has been mathematically quantified.
Here,
according to the question,
x+y+z=69
y=x+8
z=y+8
z=x+16
3x+24=69
3x=45
x=15
y=23
z=31
According to the conditions specified, piece one will be 15 inches long, piece two will be 23 inches long, and piece three will be 31 inches long. "You chop a 69-inch-long piece of wood. Three pieces of the wood are cut out. Eight inches longer than the first piece is the second one. Eight inches longer than the second piece is the third one."
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Two containers designed to hold water are side by side, both in the shape of acylinder. Container A has a diameter of 8 feet and a height bf 16 feet. Container B hasa diameter of 10 feet and a height of 8 feet. Container A is full of water and the wateris pumped into Container B until Conainter B is completely full.To the nearest tenth, what is the percent of Container A that is empty after thepumping is complete?
Okay, here we have this:
Considering the provided information, we are going to calculate what is the percent of Container A that is empty after the pumping is complete, so we obtain the following:
First we will calculate the volume of each cylinder using the following formula:
[tex]V=\pi\cdot r^2\cdot h[/tex]Applying:
[tex]\begin{gathered} V_A=\pi\cdot4^2\cdot16 \\ V_A=\pi\cdot16\cdot16 \\ V_A=256\pi \end{gathered}[/tex][tex]\begin{gathered} V_B=\pi\cdot5^2\cdot8 \\ V_B=\pi\cdot25\cdot8 \\ V_B=200\pi \end{gathered}[/tex]After pumping the water from container A to container B, the following amount remains in container A:
Remaining amount of water in A=256π-200π
Remaining amount of water in A=56π
Now, we obtain that the empty percentage that results in A is:
Empty percentage that results in A=200/256*100
Empty percentage that results in A=78.125%
Empty percentage that results in A≈78.1%
2. Factor completely
2x^2 + 8x + 6
The factors are -3 and -1
What is a Quadratic equation ?
A second-degree equation of the form ax² + bx + c = 0 is known as a quadratic equation in mathematics. Here, x is the variable, c is the constant term, and a and b are the coefficients. Since x is a second-degree variable, this quadratic equation has two roots, or solutions.
The given expression is,
2x² + 8x + 6
Put it equal to 0 so that we can solve for 'x'
2x² + 8x + 6 = 0
Now, its factors are 6x and 2x
2x² + 6x + 2x + 6 = 0
2x(x + 3) + 2(x + 3) = 0
To cross check your solution is correct or not. You've to just see the the brackets value should be same after taking common. Here the bracket value is (x+3) which is same.
(2x + 2) (x+3) = 0
split the values to solve further,
2x + 2 = 0 | x + 3 = 0
2x = -2 | x = -3
x = -2/2
x = -1
Hence, the factors are -3 and -1
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Below is the graph of a polynomial function with real coefficients. All local extrema of the function are shown in the graph.
Given
A graph of a polynomial with the real coefficients.
To find:
a) The intervals in which the function is increasing is,
[tex]\begin{gathered} (-\infty,-5) \\ (-2,2) \\ (6,\infty) \end{gathered}[/tex]b) The value of x at which the unction has local minima.
From the graph shown in the figure, there is only one local minimum at x=-2.
c) The sign of the functions leading coefficient is positive.
Since the graph is moving upwards.
d) The degree of the function is 5.
How many liters of paint must you buy to paint the walls of a rectangular prism-shaped room that is 20 m by 10 m with a ceiling height of 8 m if 1 L of paint covers40 m2? (Assume there are no doors or windows and paint comes in 1-L cans.)
17 Liters
Explanation
Step 1
find the total area to paint
we need to assume the floor wont be painted, so the total are to paint is
the are of a rectangle is gieven by:
[tex]Area=length*width[/tex]so, the total area will be
[tex]\begin{gathered} total\text{ surface area=\lparen20*10\rparen+2\lparen20*8\rparen+2\lparen10*8\rparen} \\ total\text{ surface area=200+2\lparen160\rparen+2\lparen80\rparen} \\ total\text{ surface area=200+320+160} \\ total\text{ surface area=680 m}^2 \end{gathered}[/tex]so , the area to paint is 680 square meters
Step 2
finally, to know the number of Liters need , divide the amount ( total area) by the rate of the paitn, so
[tex]\begin{gathered} paint\text{ needed=}\frac{total\text{ area}}{rate\text{ paint}} \\ paint\text{ needed=}\frac{680m^2}{40\frac{m^2}{L}}=17Liters \end{gathered}[/tex]so, the total paint needes is 17 Liters, and paint comes in 1-L cans, so
[tex]\begin{gathered} 17\text{ Liters} \\ 17\text{L}\imaginaryI\text{ters\lparen}\frac{1\text{ Can}}{1\text{ L}})=17cans \end{gathered}[/tex]therefore, the answer is
17 Liters
I hope this helps you
Which of the following are equations for the line shown below? Check all that apply. 5 (1,2) (3-6) I A. y + 6 = -4(x-3) B. y + 3 = -4(X-6) I C. y1 = -4(x-2) D. y - 2 = -4(x - 1)
We have the next points (1,2) and (3,-6)
The width of a rectangle is 6x + 8 and the length of the rectangle is 12x + 16 determine the ratio of the width to the perimeter.Supply the following:Perimeter = 21 + 2w = Ratio= w/p Final answer in simplest form:
Solution:
For this case we know that the width is given by:
w = 6x +8
The lenght is given by:
l= 12x +16
And the perimeter would be given by:
P= 2l +2w = 2(12x+16)+ 2(6x+8)= 24x+32 +12x+16=36x + 48
And then the ratio would be:
[tex]\text{ratio}=\frac{6x+8}{36x+48}=\frac{3x+4}{18x+24}[/tex]The half-life of a radioactive isotope is the time it takes for quantity of the isotope to be reduced to half its initial mass. Starting with 175 grams of a radioactive isotope, how much will be left rafter 5 half-lives? Round your answer to the nearest gram
Exponential Decay
The model for the exponential decay of a quantity Mo is:
[tex]M=M_o\cdot e^{-\lambda t}[/tex]Where λ is a constant and t is the time.
The half-life of a radioactive isotope is the time it takes to halve its initial mass. It can be calculated by making M = Mo/2 and solving for t:
[tex]\begin{gathered} \frac{M_o}{2}=M_o\cdot e^{-\lambda t} \\ \text{Simplifying:} \\ e^{-\lambda t}=\frac{1}{2} \\ \text{Taking natural log:} \\ -\lambda t=-\log 2 \\ t=\frac{\log 2}{\lambda} \end{gathered}[/tex]It's required to calculate the remaining mass of an isotope of Mo = 175 gr after 5 half-lives have passed, that is. we must calculate M when t is five times the value calculated above.
Substituting in the model:
[tex]M=175gr\cdot e^{-\lambda\cdot\frac{5\log 2}{\lambda}}[/tex]Simplifying (the value of λ cancels out):
[tex]\begin{gathered} M=175gr\cdot e^{-5\log 2} \\ \text{Calculating:} \\ M=175gr\cdot0.03125 \\ M=5.46875gr \end{gathered}[/tex]Rounding to the nearest gram, 5 grams of the radioactive isotope will be left after the required time.
Endpoint: (1,3) Midpoint: (-2,5) Please I need help ASAP
The residence of a city voted on whether to raise property taxes the ratio of yes votes to no votes was 5 to 8 if there were 4275 yes both what was the total number of votes
The ratio of votes has been given as;
[tex]Yes\colon No\Rightarrow5\colon8[/tex]This means the ratios can be expressed mathematically as;
[tex]\begin{gathered} \text{Yes}=\frac{5}{5+8}\Rightarrow\frac{5}{13} \\ No=\frac{8}{5+8}\Rightarrow\frac{8}{13} \end{gathered}[/tex]If there were 4275 YES votes, then this means the number 4275 represents 5/13.
Therefore,
[tex]\frac{5}{13}=\frac{4275}{x}[/tex]Where x represents the total number of votes. Therefore,
[tex]undefined[/tex]2. A wooden cube with volume 64 is sliced in half horizontally. The two halves are then glued together to form a rectangular solid which is not a cube. What is the surface area of this new solid? A.128 B. 112 C. 96 D. 56
we have that
the volume of the cube is equal to
V=b^3
64=b^3
b^3=4^3
b=4 unit
see the attached figure
the surface area of the new figure is equal to
SA=2B+PH
where
B is the area of the base
P is the perimeter of the base
H is the height
we have
B=4*8=32 unit2
P=2(4+8)=24 unit
H=2 unit
so
SA=2(32)+24*2
SA=64+48
SA=112 unit2
the answer is option BThe proof below shows that sin theta -sin^3 theta=sin2theta cos^2 theta/2cos theta
Given:
Given the steps of the proof of the equation
[tex]\sin\theta-\sin^3\theta=\frac{2\sin2\theta\cos^2\theta}{2\cos\theta}[/tex]Required: Expression missing on the thrd step
Explanation:
The second step is
[tex]\sin\theta-\sin^3\theta=\sin\theta(1-\sin^2\theta)\frac{2\cos\theta}{2\cos\theta}[/tex]from which leads to
[tex]\sin\theta-\sin^3\theta=\frac{(2\sin\theta\cos\theta)(1-\sin^2\theta)}{2\cos\theta}[/tex]The expression missing on the third step is
[tex]\frac{(2\sin\theta\cos\theta)(1-\sin^2\theta)}{2\cos\theta}[/tex]Option D is correct.
Final Answer:
[tex]\frac{(2\sin\theta\cos\theta)(1-\sin^2\theta)}{2\cos\theta}[/tex]find the height of the trapezoidA=80 CM2 7Cm9CM
The area formula for trapezoids is
[tex]A=\frac{(B+b)h}{2}[/tex]Where B = 9 cm, b = 7 cm, and A = 80 cm2. Let's replace these dimensions to find h
[tex]\begin{gathered} 80=\frac{(9+7)\cdot h}{2} \\ 160=16h \\ h=\frac{160}{16} \\ h=10 \end{gathered}[/tex]Hence, the height is 10 cm.Answer question number 20. The question is in the image.Reference angle is the angle form by the terminal side and the x-axis.
Answer: We have to sketch the angle and find the reference angle for the 20:
[tex]\frac{8\pi}{3}[/tex]The reference angle is an angle between the terminal side of the angle and the x-axis.
[tex]\theta_R=180^{\circ}-\theta[/tex]The provided angle is:
[tex]\begin{gathered} \theta=\frac{8\pi}{3}=480^{\circ} \\ \\ 480^{\circ}=480^{\circ}-360^{\circ}=120^{\circ} \\ \\ \theta=120^{\circ} \end{gathered}[/tex]Sketch of the angle:
Therefore the reference angle is:
[tex]\begin{gathered} \theta_R=180^{\circ}-\theta \\ \\ \theta_R=180^{\circ}-120^{\circ} \\ \\ \theta_R=60^{\circ} \end{gathered}[/tex]I need to find two sets of coordinates and graph them. Please help?!
Answer
The two coordinates on the line include
(0, -1.5) and (-4.5, 0)
The graph of the line is presented below
Explanation
We are asked to plot the grap of the given equation of a straight line.
To do that, we will obtainthe coordinates of two points on the line.
These two points will preferrably be the intercepts of the line.
y = (-x/3) - (3/2)
when x = 0
y = 0 - (3/2)
y = -(3/2)
y = -1.5
First coordinate and first point on the line is (0, -1.5)
when y = 0
0 = (-x/3) - (3/2)
(x/3) = -(3/2)
x = (-3) (3/2)
x = -(9/2)
x = -4.5
Second coordinate and second point on the line is thus (-4.5, 0)
So, to plot the line, we just mark these two points and connect them to each other.
The graph of this line is presented under 'Answer' above.
Hope this Helps!!!
May I please get help with this math problem it’s so confusing
We have to find the value of z and x.
We assume that lines g and h are parallel.
Then, z and the angle with measure 85° are consecutive interior angles.
As they are conscutive interior angles, their measures add 180°.
Then, we can write:
[tex]\begin{gathered} z+85\degree=180\degree \\ z=180-85 \\ z=95\degree \end{gathered}[/tex]Then, we can relate the angle with measure z with the angle with measure (6x-109). They are vertical angles and, therefore, they have the same measure.
Then, we can write:
[tex]\begin{gathered} z=6x-109 \\ 95=6x-109 \\ 95+109=6x \\ 204=6x \\ x=\frac{204}{6} \\ x=34 \end{gathered}[/tex]Answer: z = 95 and x = 34.
Using the Rational Roots Theorem which of the values shown are potential roots of ) = 32-132-3x + 457 Select all that apply. +1/3 +5 +5/3 +9 +1 +15 +3 +45
To solve this problem, you find the value of x that will make the function to be = 0 by substituting the likely values from the option into the eqaution and checking if after the simplification the value is 0
so checking
[tex]\begin{gathered} \text{The factors betwe}en\text{ }3\text{ and 45 are } \\ 1,3,5,9,15,45 \\ \text{factors of 3 are 1,3} \end{gathered}[/tex]we have
[tex]\begin{gathered} =3x^{^3}-13x^2-3x\text{ +45} \\ \pm1,\text{ 3, 5,9, 15,45} \\ \pm\frac{1}{3},\text{ 1, 5/3, 3, 5 , 15} \\ \text{values that apply are +3 twice and -5/3} \end{gathered}[/tex]the table shows the number of miles people in the us traveled by car annually from 1975 to 2015
In the year 2022, the predicted number of miles of travels would be 3.601 trillion miles.
What is a model?
The term model has to do with the way that we can be able to predict the interaction between variables. In this case, we can see that there is a line of best fit as we can see from the complete question which is in the image that have been attached to his answer.
The question is trying to find out the number of miles that people are going to travel in the year 2022 based on the line of best fit that have been given in the question that we have attached here.
We know that; y = 0.048x + 1.345. Recall that x here stands for the number of years that have passed since the year 1975. We now have 47 years passed since 1975 thus;
y = 0.048(47) + 1.345
y = 3.601 trillion miles
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find the unit price of a 3 pack of bottle juice for $6.75 fill in the amount per bottle of juice
EXPLANATION
Let's see the facts:
Unit price = $6.75
Number of packs = 3
The unit price is given by the following relationship:
[tex]\text{Unit price= }\frac{6.75}{3}=2.25\frac{\text{dollars}}{\text{pack}}[/tex]The unit price is 2.25 $/pack
I need help on thisChange the equation into a equivalent equation written in the Slope-intercept form. x -7y + 5 =0
The slope-intercept form is an equation as follows:
[tex]y=mx+b[/tex]Then, we need to change the original equation in this equivalent:
[tex]-7y=-5-x\Rightarrow-7y=-x-5\Rightarrow7y=x+5[/tex]Dividing the total equation by 7, we have:
[tex]\frac{7}{7}y=\frac{x}{7}+\frac{5}{7}\Rightarrow y=\frac{1}{7}x+\frac{5}{7}[/tex]Therefore, the slope-intercept form is:
[tex]y=\frac{1}{7}x+\frac{5}{7}[/tex]A test was given to a group of students. The grades and gender are summarized below A B C TotalMale 5 9 2 16Female 7 11 12 30Total 12 20 14 46If one student is chosen at random from those who took the test, find the probability that the student got a 'C' GIVEN they are female.
Probability that the student got a 'C' GIVEN they are female = number of females that got a C in the test/number of females
From the information given,
number of females that got a C in the test = 12
number of females = 30
Thus,
Probability that the student got a 'C' GIVEN they are female = 12/30
We would simplify the fraction by dividing the numerator and denominator by 6. Thus,
Probability that the student got a 'C' GIVEN they are female = 2/5
please help need answer asap
Answer:
x = 34 degrees, y = 73
Step-by-step explanation:
Since the triangle is isosceles, the base angles are congruent (equal). First, find the supplement angle by doing 180-107, which gives you 73 for the base angles, which include y. Now there is a theorem that states the 2 remote interior angles are equivalent to the exterior angle, which means 107 = 73 + x. This gives us x = 34
I hope this helps!
Use the same process for the second one.
help meee pleaseeee pleasee
Hey I need help with my homework help me find the points on the graph too please Thankyouu
Given the function:
g(x) = 3^x + 1
we are asked to plot the graph of the function.
Using the table:
x y
-2 10/9
-1 4/3
0 2
1 4
2 10
The graph:
The expomential functions have a horizontal asymptote.
The equation of the horizontal asymptote is y = 1
Horizontal Asymptote: y = 1
To find the domain is finding where the question is defined.
The range is the set of values that correspond with the domain.
Domain: (-infinity, infinity), {x|x E R}
Range: (1, infinity0, {y|y > 1}.
If a price changes from $105,300 to $104,399 will that be a percentincrease or decrease?
If the price changes from $105,300 to $104,399, It means that there is a decrease in price.
Decrease = 105,300 - 104,399 = $901
The percentage decrease is gotten by dividing the decrease by the initial price and multiplying by 100. It becomes
[tex]\frac{901}{105300}\text{ }\times\text{ 100 = 0.8557\%}[/tex]By rounding up to the nearest whole number, it becomes 1%
The percent decrease is 1%
please answer this question
Answer:
3
Step-by-step explanation:
Given expression:
[tex]\dfrac{(14^2-13^2)^{\frac{2}{3}}}{(15^2-12^2)^{\frac{1}{4}}}[/tex]
Following the order of operations, carry out the operations inside the parentheses first.
Apply the Difference of Two Square formula [tex]x^2-y^2=\left(x+y\right)\left(x-y\right)[/tex]
to the operations inside the parentheses in both the numerator and denominator:
[tex]\implies \dfrac{((14+13)(14-13))^{\frac{2}{3}}}{((15+12)(15-12))^{\frac{1}{4}}}[/tex]
Carry out the operations inside the parentheses:
[tex]\implies \dfrac{((27)(1))^{\frac{2}{3}}}{((27)(3))^{\frac{1}{4}}}[/tex]
[tex]\implies \dfrac{(27)^{\frac{2}{3}}}{(81)^{\frac{1}{4}}}[/tex]
Carry out the prime factorization of 27 and 81.
Therefore, rewrite 27 as 3³ and 81 as 3⁴:
[tex]\implies \dfrac{(3^3)^{\frac{2}{3}}}{(3^4)^{\frac{1}{4}}}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies \dfrac{3^{(3 \cdot \frac{2}{3})}}{3^{(4 \cdot \frac{1}{4})}}[/tex]
[tex]\implies \dfrac{3^2}{3^1}[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}:[/tex]
[tex]\implies 3^{(2-1)}[/tex]
[tex]\implies 3^1[/tex]
[tex]\implies 3[/tex]
Given that,
→ ((14² - 13²)^⅔)/((15² - 12²)^¼)
Evaluating the problem,
→ ((14² - 13²)^⅔)/((15² - 12²)^¼)
→ ((196 - 169)^⅔)/((225 - 144)^¼)
→ (27^⅔)/(81^¼)
→ ((3³)^⅔)/((3⁴)^¼)
→ (3²)/3
→ 9/3 = 3
Therefore, the solution is 3.
the variable w varies inversely as the cube of v. if k is the constant of variation, which equation represents this situation?a: qv^=kb: q^3 v= kc: q/v^3=kd: q^3/v=k picture listed below
Solution
Given that:
[tex]\begin{gathered} q\propto\frac{1}{v^3} \\ \\ \Rightarrow q=\frac{k}{v^3} \\ \\ \Rightarrow k=qv^3 \end{gathered}[/tex]Option A.
Teresa is participating in a 4day cross-country bike challenge. She biked it for 61, 67, and 66 miles on the first three days. How many miles does she need to bike on the last day so that her average (mean) is 63 miles per day?
The number of miles that she need to bike on the last day so that her average (mean) is 63 miles per day is 62 miles.
What is a mean?The mean is the average of a set of numbers. Let the biking of the last day be represented as x. This will be:
(61 + 67 + 66 + x) / 4 = 64
(194 + x) / 4 = 64
Cross Multiply
194 + x = 64 × 4
194 + x = 256
Collect like terms
x = 256 - 194
x = 62
The miles is 62 miles.
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Solve the following and give the interval notation of the solution and show the solution on a number line. 6x-12(3-x) is less than or equal to 9(x-4)+9x
The Solution:
The given inequality is
[tex]6x-12(3-x)\leq9(x-4)+9x[/tex]Clearing the brackets, we get
[tex]6x-36+12x\leq9x-36+9x[/tex]Collecting the like terms, we get
[tex]\begin{gathered} 6x+12x-9x-9x\leq-36+36 \\ \end{gathered}[/tex][tex]\begin{gathered} 18x-18x\leq0 \\ 0\leq0 \end{gathered}[/tex]So, the solution is true for all real values of x.
The interval notation of the solution is
[tex](-\infty,\infty)[/tex]I need help with #1 and 2 please I’m struggling
The slope of a line perpendicular to other line is the negative reciprocal of the slope.
This means, if the slope of a line is x, the slope of a perpendicular line will be:
[tex]-\frac{1}{x}[/tex]Then , the first thing we should do is to find the slope of f(x).
To find the slope of a line that passes two points P and Q we use:
[tex]\begin{gathered} \begin{cases}P=(x_p,y_p) \\ Q=(x_q,y_q)\end{cases} \\ \text{slope}=\frac{y_p-y_q}{x_p-x_q} \end{gathered}[/tex]In this case, we can use P = (1, 4) and Q = (-3, 2)
Then:
[tex]\text{slope}=\frac{4-2}{1-(-3)}=\frac{2}{4}=\frac{1}{2}[/tex]Now, we know that the slope of g(x) is perpendicular to f(x) which has a slope of 1/2
The reciprocal is:
[tex]\frac{1}{2}\Rightarrow\frac{2}{1}=2[/tex]And to make it the negative, we multiply by (-1):
[tex]2\cdot(-1)=-2[/tex]Thus, g(x) has a slope equal to -2