Given the following expression
[tex]x^2+y^2\text{ + 6x - 10y - 2 = 0}[/tex]Firstly, we need to find the perfect square
[tex]\begin{gathered} \text{Collect the like terms} \\ x^2+6x+y^2\text{ - 10y = 2} \\ \text{ Find the p}\operatorname{erf}ect\text{ square} \\ x^2\text{ + (}\frac{6}{2}x)+y^2\text{ - (}\frac{10}{2}y)\text{ = 2} \\ x^2+3x+(3)^2+y^2-5y+(-5)^2\text{ = 2} \\ x^2+3x+9+y^2\text{ - 5y + 25 = 2} \\ \text{Add p}\operatorname{erf}ect\text{ square to both sides} \\ x^2+6x+y^2\text{ - 5y = 2 + 9 + 25} \end{gathered}[/tex]Can you help me with a word problem?Under continuous compounding, the amount of time t in years required for an investment to double is a function of the interest rate r according to the formula:=tln2r(a)If you invest $3000 how long will it take the investment to reach $6000 if the interest rate is 2.5%? Round to one decimal place.(b)If you invest $3000 how long will it take the investment to reach $6000 if the interest rate is 8%? Round to one decimal place.(c)Using the doubling time found in part (b), how long would it take a $3000 investment to reach $12,000 if the interest rate is 8%? Round to one decimal place.
Answer:
a) the amount of time in years it will take for the given investment to double is;
[tex]t=27.7\text{ years}[/tex]b) the amount of time in years it will take for the given investment to double is;
[tex]t=8.7\text{ years}[/tex]Explanation:
Given that under continuous compounding, the amount of time t in years required for an investment to double is a function of the interest rate r according to the formula;
[tex]t=\frac{\ln 2}{r}[/tex]a) we want to find the amount of time it will take a $3000 investment to reach $6000 (i.e double) for an interest rate of 2.5%.
[tex]r=2.5\text{\%=}\frac{\text{2.5}}{100}=0.025[/tex]Applying the given formula;
[tex]\begin{gathered} t=\frac{\ln 2}{0.025} \\ t=27.7\text{ years} \end{gathered}[/tex]Therefore, the amount of time in years it will take for the given investment to double is;
[tex]t=27.7\text{ years}[/tex]b) we want to find the amount of time it will take a $3000 investment to reach $6000 (i.e double) for an interest rate of 8%.
[tex]r=8\text{ \%}=\frac{8}{100}=0.08[/tex]Applying the given formula;
[tex]\begin{gathered} t=\frac{\ln 2}{0.08} \\ t=8.7\text{ years} \end{gathered}[/tex]Therefore, the amount of time in years it will take for the given investment to double is;
[tex]t=8.7\text{ years}[/tex]2Louise measured the perimeter of her rectangular scrapbook to be 144 cm. If the scrapbook is 39 cm wide, how long is thescrapbook?OA. 33 cmOB. 66 cmOC. 72 cmOD. 31 cm行ResetSubmit
Given the word problem, we can deduce the following information:
1. The perimeter of her rectangular scrapbook to be 144 cm.
2. The scrapbook is 39 cm wide.
To determine the length of the scrapbook, we use the formula for the perimeter of a rectangle as shown below:
[tex]P=2(l+w)[/tex]where:
P=Perimeter =144 cm
l=length
w=width =39 cm
We plug in what we know:
[tex]\begin{gathered} P=2(l+w) \\ 144=2(l+39) \\ Simplify\text{ and rearrange} \\ \frac{144}{2}=l+39 \\ 72=l+39 \\ l=72-39 \\ Calculate \\ l=33\text{ cm} \end{gathered}[/tex]Therefore, the answer is:
OA. 33 cm
On March 10th, the owner of The Granary borrowed $80,000 on a 180-day promissory note at 10.5% interest. Find the maturity value on the note
SOLUTION
Notes are often a key component of how a business finances its operations. In general note are a short-term commercial financing.
Write out the information given
[tex]\begin{gathered} \text{ Principal=\$80,000} \\ \text{ Rate=10.5 \%}=\frac{10.5}{100}=0.105 \\ \\ \text{Time}=\frac{180}{360}=0.5 \end{gathered}[/tex]The maturity value formula is given by
[tex]\text{maturity value= Principal(1+rate x time)}[/tex]Substitute, tyhe value into the formula, we have
[tex]\text{Maturity value =80000(1+(0.5x0.105))}[/tex]Then
[tex]\begin{gathered} \text{Maturity value=80 000(1+0.0525)} \\ \text{Maturity value=80 000(1.0525)} \end{gathered}[/tex]Hence
[tex]\text{Maturity value=}84\text{ 200}[/tex]Hence
The maturity value on the note will be $84 200
Answer: $84 200
simplify and CLT: 4(7y-3) +3y +11 *
First you multiply the 4 ( 7y - 3 )
[tex]4(7y-3)=28y-12[/tex]Then you get:
[tex]28y-12+3y+11[/tex]you make the opperations (add or substract)
[tex]31y-1[/tex]And the final expression is : 31 y - 1Write the inequality in slope-intercept form\-10x + 2y > 14
Slope-intercept form is given as:
y = mx + b
we need to rewrite our inequality in a way y represents the subject of formula:
[tex]\begin{gathered} -10x\text{ + 2y > 14} \\ \text{Add 10x to both sides:} \\ -10x\text{ + 10x + 2y > 14 + 10x} \\ 2y\text{ }>\text{ 14 + 10x} \end{gathered}[/tex][tex]\begin{gathered} 2y\text{ > 10x +14} \\ \text{divide both sides by 2:} \\ \frac{2y}{2}\text{ > }\frac{10x}{2}\text{ +}\frac{14}{2} \\ y\text{ > 5x + }7 \end{gathered}[/tex]Some students developed math games for a school carnival. One of the games uses the spinner shown below. The outcomes for the spinner are for a player to win either 1 coupon, 5 coupons, or 10 coupons. It costs a player 3 coupons to turn the spinner once. What is the expected payoff, in coupons, for the game? A. 1/8 coupon B. 5/8 coupon C. 3 1/8 coupons D. 3 5/8 coupons
The expected value is fiven as:
[tex]EV=\sum ^{}_{}x_iP(x_i)[/tex]where xi is the possible outcome and P(xi) is its probability.
From the spinner we notice that:
[tex]\begin{gathered} P(1)=\frac{4}{8}=\frac{1}{2} \\ P(5)=\frac{3}{8} \\ P(10)=\frac{1}{8} \end{gathered}[/tex]Then the expected value is:
[tex]\begin{gathered} EV=1(\frac{4}{8})+5(\frac{3}{8})+10(\frac{1}{8}) \\ EV=\frac{4}{8}+\frac{15}{8}+\frac{10}{8} \\ EV=\frac{29}{8} \\ EV=3\frac{5}{8} \end{gathered}[/tex]Therefore the expected value is 3 5/8 and the answer is D.
I need help with graphing the points and solving it
We have a system of inequalities:
[tex]\begin{gathered} y\ge-2x-2 \\ y>\frac{1}{3}x \end{gathered}[/tex]We can graph this inequalities as:
The points that satisfy both equations should be in the most shaded area (area marked with A)
If we graph the points listed, we get:
The point (-1,4) is the only one that satisfy both inequalities.
Answer: Point (-1,4) [Option B]
The area of a parking lot is 300 square meters. A bicycle requires 3 square meters. A bus requires 9 square meters. The parking lot can hold only 40 vehicles. If a bicycle is charged $1.50 and a bus $3, find the possible income region.
From the question given,
Area of the parking lot = 300 square meters
Bicycle requirement = 3 square meters
Bus requirement = 9 square meters
Total number of vehicle = 40 vehicles
Let x be the number of bicycles and y be the number of bus
x + y = 40 -------------------------(1)
3x + 9y = 300 ---------------------------(2)
We can solve by elimination method
Multiply through equation (1) by 3 in other to eliminate x and then we will solve for y
3x + 3y = 180----------------------------(3)
Equation (2) - Equation (3)
6y = 180
Divide both-side by 6
y = 30
substitute y=30 into equation (1) and solve for x
x + 30 = 40
subtract 30 from both-side
x = 40 -30
x=10
Number of bicycle is 10 and number of bus is 30
Since, a bicycle is charged $1.50 and bus is $3
Then the total income is;
$(10 × 1.50 + 30 × 3)
= $(15 + 90)
= $105
Therefore, the the possible income region is $105.
Graph the following function by considering how the function x^2 has been shifted, reflected, stretched or compressed.
Given the quadratic function g(x) defined as:
[tex]g(x)=-\frac{(x-2)^2}{2}-2[/tex]We can go from the function f(x) = x² to g(x) making the transformations:
1) A reflection about the x-axis:
[tex]f(x)\to-f(x)[/tex]2) A horizontal dilation by a factor of 1/2:
[tex]f(x)\to\frac{1}{2}f(x)[/tex]3) A shift of 2 units down:
[tex]f(x)\to f(x)-2[/tex]4) A shift of 2 units right:
[tex]f(x)\to f(x-2)[/tex]Combining all these transformations:
[tex]f(x)\to-\frac{f(x-2)}{2}-2=-\frac{(x-2)^2}{2}-2=g(x)[/tex]Then, the graphs of f(x) (red) and g(x) (green) are:
Choose the equation below which would be the line of best fit for the scatterplot. Is the correct answer a, B, C, or D?
Solution:
To find the line of best fit for the scatterplot, we pick points from the graph as shown below
From the graph,
[tex]\begin{gathered} (x_1,y_1)\Rightarrow(2,8) \\ (x_2,y_2)\Rightarrow(1,9) \end{gathered}[/tex]To find the equation of a straight line, the formula is
[tex]\frac{y-y_1}{x-x_1}=\frac{y_2-y_1}{x_2-x_1}[/tex]Substitute the coordinates into the formula above
[tex]\begin{gathered} \frac{y-8}{x-2}=\frac{9-8}{1-2} \\ \frac{y-8}{x-2}=\frac{1}{-1} \\ \frac{y-8}{x-2}=\frac{-1}{1} \\ Crossmultiply \\ 1(y-8)=-1(x-2) \\ y-8=-x+2 \\ y=-x+2+8 \\ y=-x+10 \end{gathered}[/tex]Hence, the line of best fit is
[tex]y=-x+10[/tex]what is the difference in the length between the shortest
1) Considering the group of the longest worms size is 1 and 1/2 inch and the shortest ones have the size of 3/4 of an inch.
2) Firstly, let's convert the mixed number to an improper fraction and then subtract.
[tex]1\frac{1}{2}=\frac{(2\cdot1+1)}{2}=\frac{3}{2}[/tex]We do this by conserving the denominator and multiplying it by the whole number and adding to the numerator:
To find the Least Common Multiple (LCM), we proceed this way:
We divide those numbers for a prime number, that divide both (2 and 4) and go on until we reach 1. Then we multiply the numbers on the right. ( 2 and 2)
3) Now let's subtract then. Since they have different denominators, then we must calculate the Least Common Multiple of both:
[tex]\begin{gathered} \frac{3}{2}-\frac{3}{4} \\ \frac{(4\colon2\times3)-(4\colon4\times3)}{4}= \\ \frac{6-3}{4}= \\ \frac{3}{4} \end{gathered}[/tex]The LCM between 2,4 is 8. So the difference between the longest size (3/2 inches or 1 1/2 ) and the shortest 3/4 is equal to 3/4
I need help pleaseThe Circumference of the bottom of the barrel is53.40703How high must Mario jump to get over the barrel?
SOLUTION:
Case: Circumference
Method:
Circumference, C= 53.40703
The height of the barrel described here is the diameter. Mario would have to jump a little above the diameter of the barrel but not too high to get to the fire.
The formula connecting the Circumference of a circle to its diameter is given as:
[tex]\begin{gathered} C=\pi d \\ 53.40703=\pi d \\ d=\frac{53.40703}{\pi} \\ d=16.99998564071 \end{gathered}[/tex]Final answer:
Mario has just jump a little above 16.99998564071 units high
Find the equations of any horizontal asymptotes. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
ANSWER
OPTION B
The horizontal asymptote is 7
STEP-BY-STEP EXPLANATION
Given information
[tex]f(x)\text{ = }\frac{7|x|}{x\text{ + 2}}[/tex]Therefore, the correct option is B
Part B
The horizontal asymptote of a rational function can be determined by looking at the degree of numerator and denominator.
Therefore, the horizontal asymptote is 7
Sta4.A radar detector records the speeds of a group of motorists that pass by. If the mean is 52 mph and thestandard deviation is 3.2 mph, find the Z-score for 48 mph.Answer:
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
mean = 52 mph
standard deviation = 3.2 mph
score = 48 mph
Step 02:
Z-score:
[tex]z=\text{ }\frac{x-u}{\sigma}[/tex][tex]z=\frac{52-48}{3.2}=\frac{4}{3.2}=1.25[/tex]The answer is:
z = 1.25
I don't understand how to do this question. the equation is [tex]g(x) = \pi \sin( \frac{x}{2} ) + \pi[/tex]
We have the next function
[tex]g(x)=\pi\sin (\frac{x}{2})+\pi[/tex]the amplitude is π
The natural frequency is
[tex]\omega=\frac{1}{2}[/tex]with this, we can calculate the frequency and the period
[tex]\omega=2\pi f=\frac{2\pi}{T}[/tex]for frequency
[tex]f=\frac{\omega}{2\pi}=\frac{\frac{1}{2}}{2\pi}=\frac{1}{4\pi}[/tex]the period is
[tex]T=\frac{1}{f}=4\pi[/tex]The phase shift is 0
the vertical translation is π
the equation of midline is
[tex]y=\pi[/tex]The graph of the function is
where g(x) is the graph in red and the midline is the graph in blue
Find the length of the radius Round to the nearest whole number
Let's picture the situation described by the exercise:
We have a circle with radius r, and a (green) sector with an area of 361.6 m^2, and with a central angle of 288 degrees. To solve this exercise, we merely need to remember the formula for the area (AS) of a sector:
[tex]AS=\frac{\text{angle}}{360}\cdot\pi\cdot r^2.[/tex]For we are looking for the radius, we need to solve this "equation" for the variable r:
[tex]\begin{gathered} AS=\frac{\text{angle}}{360}\cdot\pi\cdot r^2, \\ 360\cdot AS=(\text{angle)}\cdot\pi\cdot r^2, \\ \frac{360\cdot AS}{(\text{angle)}\cdot\pi}=r^2, \\ r^2=\frac{360\cdot AS}{(\text{angle)}\cdot\pi}, \\ r=\sqrt[]{\frac{360\cdot AS}{(\text{angle)}\cdot\pi}}\text{.} \end{gathered}[/tex]Evaluating for the values of AS and "angle" of our sector, we get
[tex]\begin{gathered} r=\sqrt[]{\frac{360\cdot AS}{(\text{angle)}\cdot\pi}}\leftarrow\begin{cases}AS=361.6m^2 \\ angle=288 \\ \pi=3.14\end{cases}, \\ r=\sqrt[]{\frac{360\cdot361.6m^2}{288\cdot(3.14)}}, \\ r\approx\sqrt[]{143.9m^2}, \\ r\approx11.9m. \end{gathered}[/tex]Finally, we must round up the value of r we just obtained. Note that the first decimal place (9) of 11.9 is greater than 5, then by the rounding rule we must add one tenth, to get
[tex]r=12m[/tex]AnswerThe radius of the circle is
[tex]r=12m[/tex]- 3х + 5 + 8x - 2 Show your work answer
- 3х + 5 + 8x - 2
Add like terms:
(-3x + 8x) + (5 - 2)
(5x) + (3)
5x + 3
fill in the blank for box and dox plot mathimatics
the answer is
the range. , the median and the quartiles
Please help me with the question below(also please answer the question in a maximum of 5-10 minutes and if you can please give me an example).
Remember that
In any right triangle, opposite angles are complementary
that means
in this problem
x+55=90 degrees
x=90-55
x=35 degreesUse technology to find points and then graph the line y=-5(x-5)+1, following the instructions below.
According to the given question, we are to plot the graph of the line y= -5(x-5)+1. To do that, we will first create coordinate points on the plane in the form of (x, y)
If y = 10, then;
10 = -5(x-5) + 1
10-1 = -5(x-5)
9 = -5(x - 5)
x- 5 = -9/5
x - 5 = -1.8
x = -1.8 + 5
x = 3.2
Hence one of the coordinates that fit is ABB(3.2, 10)
If y = 5, then;
5 = -5(x-5) + 1
5 - 1 = -5(x - 5)
4 = -5(x - 5)
x - 5 = -4/5
x - 5 = -0.8
x = -0.8 + 5
x = 4.2
Another coordinate point that fit into the plane is B(4.2, 5)
The required graph is as plotted below;
i inserted a picture of the questions, you have to click on the picture to fully see all the questions
(14)
[tex]\begin{gathered} 1\text{ minute=60 second} \\ 4.4\times1\text{ minute=4.4}\times60\text{ second} \\ 4.4\text{ minute =264 sec.} \end{gathered}[/tex](15)
[tex]\begin{gathered} 1\text{ kelometer =1000meter} \\ 2.7\times1\text{ kilometers=2.7}\times1000meters \\ =2700\text{ meter} \end{gathered}[/tex](16)
[tex]\begin{gathered} 1\text{ pound=16 ounce} \\ 2.6\text{pound}=41.6\text{ounces} \end{gathered}[/tex](17)
[tex]\begin{gathered} 1\text{ hours=60 min.} \\ 7.1\text{ hours=426 min.} \end{gathered}[/tex](18)
[tex]\begin{gathered} 1\text{ inch=2.54 cm} \\ 5\text{inch}=12.7\operatorname{cm} \end{gathered}[/tex](19)
[tex]\begin{gathered} 1\text{ pound=16 ounce} \\ 1\text{ ounce=}\frac{1}{16}pound \\ 120\text{ ounce=120}\times\frac{1}{16}pound \\ =7.5\text{ pound} \end{gathered}[/tex](20)
[tex]\begin{gathered} 1\text{ f}eet=12\text{ inch} \\ 7.5\text{ f}eet=90\text{ inch} \end{gathered}[/tex]the cost of 2 pencils and 3 notebooks is 3.35.the cost of 4 pencils and 2 notebooks is 3.70. How much is the cost of one pencil
Answer
Explanation
Let the cost of 1 pencil be p
Let the cost of 1 notebook be n
The cost of 2 pencils and 3 notebooks is 3.35
2p + 3n = 3.35
The cost of 4 pencils and 2 notebooks is 3.70
4p + 2n = 3.70
We can then combine the two equations and solve simultaneously
2p + 3n = 3.35
4p + 2n = 3.70
Multiply the first equation by 2 and the second equation by 1
(2p + 3n = 3.35) × 2
(4p + 2n = 3.70) × 1
276 fewer than h equals 92 Type a slash (/) if you want to use a division sign.
The question can be written as;
h-276 = 92
22 is 0.2% of what number? Use pencil and paper. Would you expect the answer to be a lot less than22, slightly less than 22, slightly greater than 22, or a lot greater than 22? Explain.22 is 0.2% of |(Type an integer or a decimal.)
We will investigate how to express percentages as decimals ( fractions ). Then we will see how to manipulate these percentages to get the starting or resltant value.
We are asked to determine the " starting number " whose 0.2% would result in the final number:
[tex]22[/tex]We will first see how to deal with the percentage ( 0.2% ) and the starting number which is an unknown. So lets suppose:
[tex]\text{Starting number = x}[/tex]Then we will see how to deal with percentages. The way we deal is by using a basic mathematical operator of ( multiplication ) and convert the percentage into a fraction as follows:
[tex]\begin{gathered} \text{Starting number }\cdot\text{ Percentage} \\ x\cdot\frac{0.2}{100} \end{gathered}[/tex]Then we will equate the above expression to the resultant as follows:
[tex]x\cdot\frac{0.2}{100}\text{ = 22}[/tex]Now we have aa full fledged equation. We can easily manipulate and solve for the unknown number ( x ) as follows:
[tex]\begin{gathered} x\cdot0.2\text{ = 2200} \\ x\text{ = }\frac{2200}{0.2} \\ \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ = 11,000}} \end{gathered}[/tex]We see the unknown starting number ( x ) is very large as should have been the case. This is due to the fact that ( 0.2% ) is a very small proportion of a number such that it results in a two-digit number like ( 22 ). If the percentage is low but resultant is high, then the starting number must be very large!
Answer:
[tex]\textcolor{#FF7968}{11,000}[/tex]Shown below are three scatter plots and their lines of best fit.Also shown are three residual plots.Answer the questions that follow.Option second question:Perfect modelGood (but not perfect) modelNot an appropriate model
Answer:
Residual Plot 1 corresponds to Scatter plot C - What best describes the line of best fit: Perfect Model
Residual Plot 2 corresponds to Scatter Plot B - What best describes the line of best fit: Good (but not perfect) model
Residual Plot 3 corresponds to Scatter Plot A - What best describes the line of best fit: Not an appropriate model
Explanation:
The Residual Plot shows the difference of each data point with the line of best fit.
In the Residual Plot 1, the distance between the data point and the line of best fit is 0, thus the corresponding scatter plot is one where all the points are on the line of best fit. This is Scatter Plot C. Since the line perfectly describes the data, we call it a perfect model.
In the Resig
The are of a square is 36 square centimeters. What is the length of each side of the sqaure
We are given that the area of a square is 36 square centimeters. The area of a square is given by the following formula:
[tex]A=l^2[/tex]Since we are given the area we can replace in the formula:
[tex]36=l^2[/tex]Now we can solve for the length by taking the square root to both sides:
[tex]\sqrt[]{36}=\sqrt{l^2}[/tex]Solving the operations we get:
[tex]6=l[/tex]Therefore, the length of the square is 6 cen
identify the solution for the following graph of a system of equation
The solution of a system of equations is given graphically by the intercept between the lines. If the lines are parallel, the system has no solution, if the lines are equal, the system has infinite solutions, and if the lines intercept each other once, this point represents the solution.
In our graph, we have an interception at (-1, 3), which is the solution for our system.
I need help to find lcd in rational equation here's a example 1/12;5/18;7/24
So we need to find the LCM of these denominators (12,18,24)
The LCM will be 2x3x2x2x3 = 72
Suppose the sun casts a shadow off a 20-footbuilding. If the angle of elevation to the sun is 65°,how long is the shadow to the nearest tenth of afoot?20 feet6509.3 feet13.2 feet17.2 feet21.1 feet
The triangle formed by the system in the question is shown below:
The length of the shadow is represented by x.
We can use the Tangent Trigonometric ratio to solve for x.
The ratio is given as
[tex]\tan \theta=\frac{\text{opp}}{\text{adj}}[/tex]From the diagram above, we have the following parameters:
[tex]\begin{gathered} \theta=65 \\ \text{opp }=20 \\ \text{adj }=x \end{gathered}[/tex]Hence, we can substitute as
[tex]\tan 65=\frac{20}{x}[/tex]Solving for x, we have
[tex]\begin{gathered} 2.145=\frac{20}{x} \\ \therefore \\ x=\frac{20}{2.145} \\ x=9.3\text{ feet} \end{gathered}[/tex]The correct answer is the FIRST OPTION (9.3 feet).
solve each system of equationsy=-x + 16y=-x - 8
Solving the simultaneous equation using the substitution method,
[tex]\begin{gathered} y=x+16\text{ equation (1)} \\ y=x-8\text{ equation (2)} \end{gathered}[/tex]Substituting equation (2) into equation (1),
Replace y with x-8 in equation (1),
[tex]\begin{gathered} x-8=x+16 \\ x-x=16+0 \\ 0=16 \\ \text{Therefore, no solution.} \end{gathered}[/tex]Hence, the question has no solution.