The width of a rectangle is 6 less than twice its length. If the area of the rectangle is 170 cm2 , what is the length of the diagonal?The length of the diagonal is cm.Give your answer to 2 decimal places.Submit QuestionQuestion 25
Answers
The formula to find the area of a rectangle is:
[tex]\begin{gathered} A=l\cdot w \\ \text{ Where} \\ \text{ A is the area} \\ l\text{ is the length} \\ w\text{ is the width} \end{gathered}[/tex]Since the rectangle area is 170cm², we can write the following equation.
[tex]170=l\cdot w\Rightarrow\text{ Equation 1}[/tex]On the other hand, we know that the width of the rectangle is 6 less than twice its length. Then, we can write another equation.
[tex]\begin{gathered} w=2l-6\Rightarrow\text{ Equation 2} \\ \text{ Because} \\ 2l\Rightarrow\text{ Twice length} \\ 2l-6\Rightarrow\text{ 6 less than twice length} \end{gathered}[/tex]Now, we solve the found system of equations.
[tex]\begin{cases}170=l\cdot w\Rightarrow\text{ Equation 1} \\ w=2l-6\Rightarrow\text{ Equation 2}\end{cases}[/tex]For this, we can use the substitution method.
Step 1: we replace the value of w from Equation 2 into Equation 1. Then, we solve for l.
[tex]\begin{gathered} 170=l(2l-6) \\ \text{Apply the distributive property} \\ 170=l\cdot2l-l\cdot6 \\ 170=2l^2-6l \\ \text{ Subtract 170 from both sides} \\ 0=2l^2-6l-170 \end{gathered}[/tex]We can use the quadratic formula to solve the above equation.
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}\Rightarrow\text{ Quadratic formula} \\ \text{ For }ax^2+bx+c=0 \end{gathered}[/tex]Then, we have:
[tex]\begin{gathered} a=2 \\ b=-6 \\ c=-170 \\ l=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ l=\frac{-(-6)\pm\sqrt[]{(-6)^2-4(2)(-170)}}{2(2)} \\ l=\frac{6\pm\sqrt[]{1396}}{4} \\ \end{gathered}[/tex]There are two solutions for l.
[tex]\begin{gathered} l_1=\frac{6+\sqrt[]{1396}}{4}\approx10.84 \\ l_2=\frac{6-\sqrt[]{1396}}{4}\approx-7.84 \\ \text{ The symbol }\approx\text{ is read 'approximately'.} \end{gathered}[/tex]Since the value of l can not be negative, the value of l is 10.84.
Step 2: We replace the value of l into any of the equations of the system to find the value of w. For example, in Equation 1.
[tex]\begin{gathered} 170=l\cdot w\Rightarrow\text{ Equation 1} \\ 170=10.84\cdot w \\ \text{ Divide by 10.84 from both sides} \\ \frac{170}{10.84}=\frac{10.84\cdot w}{10.84} \\ 15.68\approx w \end{gathered}[/tex]Now, the long side, the wide side and the diagonal of the rectangle form a right triangle.
Then, we can use the Pythagorean theorem formula to find the length of the diagonal.
[tex]\begin{gathered} a^2+b^2=c^2 \\ \text{ Where} \\ a\text{ and }b\text{ are the legs} \\ c\text{ is the hypotenuse} \end{gathered}[/tex]In this case, we have:
[tex]\begin{gathered} a=10.84 \\ b=15.68 \\ a^2+b^2=c^2 \\ (10.84)^2+(15.68)^2=c^2 \\ 117.51+245.86=c^2 \\ 363.37=c^2 \\ \text{ Apply square root to both sides of the equation} \\ \sqrt[]{363.37}=\sqrt[]{c^2} \\ 19.06=c \end{gathered}[/tex]Therefore, the length of the diagonal of the given rectangle is 19.06 cm rounded to 2 decimal places.
Related Questions
what is the constant of proportionality in this proportional relationship? x 2 2-1/2 3 3-1/2 y 5/2 25/8 15/4 35/8. answer choices 4/5, 5/4, 4, 5
Answers
a proportional relationship has the following form:
yyy=
Find the variance for the set of data: 22, 26, 17, 20, 20.The variance is
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The variance of a given data set with size N is given by the formula:
[tex]\begin{gathered} \sigma=\sqrt{\frac{1}{N}\sum_{i=1}^N(x_i-\mu)^2} \\ Var(X)=\sigma^2 \end{gathered}[/tex]Then, for the data set {22, 26, 17, 20, 20} and N = 5, we have:
[tex]\begin{gathered} \mu=\frac{22+26+17+20+20}{5}=21 \\ \sigma=\sqrt{\frac{1^2+5^2+(-4)^2+(-1)^2+(-1)^2}{5}}=\sqrt{\frac{44}{5}}=2\sqrt{\frac{11}{5}} \\ \therefore Var(X)=\frac{44}{5}=8.8 \end{gathered}[/tex]State the number of complex zeros and the possible number of real and imaginary zeros for each function. Then find all zeros. show all work
Answers
We have a cubic function
[tex]f(x)=x^3-3x^2-47x-87[/tex]One way to find all the zeros is by factoring, we can easily find the first zero using the divisors test if we have an independent term, at our case it's -87, one of the divisors may be a zero. The divisors of -87 is 1, 3, 29 and 87.
If we check for all of the divisors we will see that -3 is a zero. (Check with both signals).
If -3 is a zero, the D'Alembert theorem tells us that f(x) is divisible by (x+3), if we do that division we'll have a quadratic function where we can just apply the quadratic formula, then
There's a theorem that says that, if f(a) is a zero, i.e f(a) = 0, and f(x) is a polynomial, then f(x) is divisible by (x-a), in other words, we can divide f(x) by (x-a) and the rest of the division will be 0.
Therefore, let's divide our function by (x+3)
Then we can write our function as
[tex]f(x)=(x+3)(x^2-6x-29)[/tex]Look that now we have a quadratic function, and we can easily find its zeros, applying the quadratic formula
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]We have a = 1, b = -6 and c = -29. Then
[tex]\begin{gathered} x=\frac{6\pm\sqrt[]{36+4\cdot29}}{2} \\ \\ x=\frac{6\pm\sqrt[]{156}}{2} \\ \\ x=\frac{6\pm2\sqrt[]{38}}{2} \\ \\ x=3\pm\sqrt[]{38} \end{gathered}[/tex]Now we have all the zeros of f(x), it's
[tex]\begin{gathered} x=-3 \\ \\ x=3+\sqrt[]{38} \\ \\ x=3-\sqrt[]{38} \end{gathered}[/tex]As we can see there's no complex zero, all the zeros are real numbers.
The max number of complex zeros is 2 because the complex zeros always come in pairs, so if we have 1 complex zero, automatically we have another, for a 3-degree equation, there's a maximum of 2 complex zeros and 1 real zero, or all the of them are real.
Then the correct answer is A)
You choose a marble from the bag. What is the probability you will NOT choose blue?1/25/72/72
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Given a sample and required to get the probability of a particular outcome, we make a couple of considerations including:
- Sample Space: The universal set
- Required Outcome
We can identify these variables as:
Sample space: total number of marbles = 7
Required outcome: Not blue = 7 - 2 = 5
Probability is given as:
[tex]\begin{gathered} P=\text{ }\frac{\text{number of required outcome}}{Sample\text{ space}}=\frac{5}{7} \\ P=\frac{5}{7} \end{gathered}[/tex]The perimeter of the triangle below is 91 units. Find the length of the side QR. write your answer without variables.
Answers
Given:
The perimeter of the triangle, P=91.
The sides of the triangle are,
PR=4z
QR=z+3
PQ=5z-2.
The perimeter of the triangle can be expressed as,
[tex]\begin{gathered} P=PR+QR+PQ \\ P=4z+z+3+5z-2 \\ P=10z+1 \end{gathered}[/tex]Now, put P=91 in the above equation to find the value of z.
[tex]\begin{gathered} 91=10z+1 \\ 91-1=10z \\ 90=10z \\ \frac{90}{10}=z \\ 9=z \end{gathered}[/tex]Now, the length of the side QR can be calculated as,
[tex]\begin{gathered} QR=z+3 \\ QR=9+3 \\ QR=12 \end{gathered}[/tex]Now, the length of QR is 12 units.
What happens to F(x) when x is negative but approaches zero for the functionF(x) = 1/x, whose graph is shown below?
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Given: The graph of the function below
[tex]F(x)=\frac{1}{x}[/tex]To Determine: What happens to F(x) when x is negative but approaches zero
Solution:
It can be observed from the given graph that when x is negative but approaches zero, F(x) approaches negative infinity
This is as shown below
From the options provided, the best answer is F(x) is a negative number, OPTION C
Sally Sue had spent all day preparing for the prom. All the glitz and the glamour of the evening fell apart as she stepped out of the limousine and her heel broke and she fell to the ground. Within minutes, news of her crashing fall had spread to the 550 people already at the prom. The function, p(t) = 550(1-e^-0.039t) where t represents the number of minutes after the fall, models the number of people who were already at the prom who heard the news.How many minutes does it take before all 550 people already at the prom hear the news ofthe great fall? Show your work.
Answers
We have the function
[tex]p(t)=550(1-e^{-0.039t})[/tex]Therefore we want to determine when we have
[tex]p(t_0)=550[/tex]It means that the term
[tex]e^{-0.039t}[/tex]Must go to zero, then let's forget the rest of the function for a sec and focus only on this term
[tex]e^{-0.039t}\rightarrow0[/tex]But for which value of t? When we have a decreasing exponential, it's interesting to input values that are multiples of the exponential coefficient, if we have 0.039 in the exponential, let's define that
[tex]\alpha=\frac{1}{0.039}[/tex]The inverse of the number, but why do that? look what happens when we do t = α
[tex]e^{-0.039t}\Rightarrow e^{-0.039\alpha}\Rightarrow e^{-1}=\frac{1}{e}[/tex]And when t = 2α
[tex]e^{-0.039t}\Rightarrow e^{-0.039\cdot2\alpha}\Rightarrow e^{-2}=\frac{1}{e^2}[/tex]We can write it in terms of e only.
And we can find for which value of α we have a small value that satisfies
[tex]e^{-0.039t}\approx0[/tex]Only using powers of e
Let's write some inverse powers of e:
[tex]\begin{gathered} \frac{1}{e}=0.368 \\ \\ \frac{1}{e^2}=0.135 \\ \\ \frac{1}{e^3}=0.05 \\ \\ \frac{1}{e^4}=0.02 \\ \\ \frac{1}{e^5}=0.006 \end{gathered}[/tex]See that at t = 5α we have a small value already, then if we input p(5α) we can get
[tex]\begin{gathered} p(5\alpha)=550(1-e^{-0.039\cdot5\alpha}) \\ \\ p(5\alpha)=550(1-0.006) \\ \\ p(5\alpha)=550(1-0.006) \\ \\ p(5\alpha)=550\cdot0.994 \\ \\ p(5\alpha)\approx547 \end{gathered}[/tex]That's already very close to 550, if we want a better approximation we can use t = 8α, which will result in 549.81, which is basically 550.
Therefore, we can use t = 5α and say that 3 people are not important for our case, and say that it's basically 550, or use t = 8α and get a very close value.
In both cases, the decimal answers would be
[tex]\begin{gathered} 5\alpha=\frac{5}{0.039}=128.2\text{ minutes (good approx)} \\ \\ 8\alpha=\frac{8}{0.039}=205.13\text{ minutes (even better approx)} \end{gathered}[/tex]For the compound inequalities below (5-7), determine whether the inequality results in an overlapping region or a combined region. Then determine whether the circles are open are closed. Finally, graph the compound inequality. Simplify if needed. x-1>_5 and 2x<14
Answers
The inequalities are:
[tex]x-1\ge5\text{ and }2x<14[/tex]So, we need to solve for x on both inequalities as:
[tex]\begin{gathered} x-1\ge5 \\ x-1+1\ge5+1 \\ x\ge6 \end{gathered}[/tex][tex]\begin{gathered} 2x<14 \\ \frac{2x}{2}<\frac{14}{2} \\ x<7 \end{gathered}[/tex]Now, we can model the inequalities as:
So, the region that results is an overlapping region and it is written as:
6 ≤ x < 7
So, the lower limit 6 is closed and the upper limit 7 is open.
Answer: The region is overlaping and it is 6 ≤ x < 7
Rearrange the formula y = a-bx² to make x the subject.
Answers
Answer:
x = ± [tex]\sqrt{\frac{a-y}{b} }[/tex]
Step-by-step explanation:
y = a - bx² ( subtract a from both sides )
y - a = - bx² ( multiply through by - 1 )
bx² = a - y ( divide both sides by b )
x² = [tex]\frac{a-y}{b}[/tex] ( take square root of both sides )
x = ± [tex]\sqrt{\frac{a-y}{b} }[/tex]
Choose an equation that models the verbal scenario. The cost of a phone call is 7 cents to connect and an additional 6 cents per minute (m).
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"The cost of a phone call is 7 cents to connect and an additional 6 cents per minute (m)"
If "C" indicates the total cost of a phone call and "m" corresponds to the number of minutes the phone call lasted.
The phone call costs 7 cents to connect, this means that regardless of the duration of the call, you will always pay this fee. This value corresponds to the y-intercept of the equation.
Then, the phone call costs 6 cents per minute, you can express this as "6m"
The total cost of the call can be calculated by adding the cost per minute and the fixed cost:
[tex]C=6m+7[/tex]Slove this equation 19=n+22
Answers
Step-by-step explanation:
I think it helps you
please mark me as brainlist
Answer:
greyehahhsh[tex]5.5723[/tex]Which of the following is not a correct way to name the plane.
Answers
For this case the first option is correct Plane P
What is the value of 3/8 dividend by 9/10
A) 3
B 5/12
C 27/80
D 2/3
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Answer:
B 5/12 (im stupi d)
Step-by-step explanation:
(3/8)/(9/10) = (3/8) * (10/9) = 5/12
Answer:
B) [tex]\frac{5}{12}[/tex]
Step-by-step explanation:
Apply the fractions rule a/b ÷c/b = a/b × d/c
= 3/8 x 10/9
Multiply fractions a/b x c/d = [tex]\frac{axc}{b x d}[/tex]
Multiply the numbers: 3 x 10 = 30
= 3/10 8 x 9
Multiply the numbers: 8 x 9 = 72
= 30/72
Cancel the common factor: 6
5/12
7(x+2)=
4(x+4)=
9(x+6)=
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16
54
Multiply each term in the parentheses by 9 and you should get something like this: 9x+9x6 and then multiply the numbers and you should get 9x+54
hey there mr or ms could you please help me out here?
Answers
The two triangles have a common side, RQ.
Also, given the two sides (left and right) are equal.
Also, the angle between the two sides (one side given and bottom side) is given as 90 degrees.
Thus,
we have
2 sides AND 1 angle congruent in each triangle
That is:
Side-Angle-Side, which is
SAS
THe triangles are congruent according to SAS, option B
The circle graph shows how the annual budget for a company is divided by department. If the amount budgeted for support, sales, and media combined is $25,000,000, what is the total annual budget?
Answers
Answer: $50,000,000
Explanation:
First, we add up the percentage of support, sales, and media covers. Given that:
Support = 23%
Sales = 22%
Media = 5%
The total percentage would be
[tex]23\%+22\%+5\%=50\%[/tex]This would mean that $25,000,000 covers half of the annual budget. The other half would be of the same amount, therefore, the total annual budget would be:
[tex]\begin{gathered} 50\%+50\%=100\% \\ \$25,000,000+\$25,000,000=\$50,000,000 \end{gathered}[/tex]The triangles are similar, solve for the question mark. A Z с ? 15 10 12 B X D E 8 8 18 12.5 0 24
Answers
Answer:
18
Explanation:
The triangles are similar if their sides are proportional. It meant that the ratio of AB to CD is equal to the ratio of AE to CE, so we can write the following equation:
[tex]\begin{gathered} \frac{AB}{CD}=\frac{AE}{CE} \\ \frac{15}{10}=\frac{AE}{12} \end{gathered}[/tex]So, we can solve for AE as:
[tex]\begin{gathered} \frac{15}{10}\cdot12=\frac{AE}{12}\cdot12 \\ 18=AE \end{gathered}[/tex]Therefore, the measure of AE is 18
What is the equation of the line below in slope-intercept form?(4 Points)x-3y = 6y =- 2y = 3x - 2y = - ** - 2y = -3x - 2
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Let's make y the subject of the equation.
[tex]\begin{gathered} x-6=3y \\ y=\frac{x-6}{3} \\ y=\frac{1}{3}x-\frac{6}{3} \\ y=\frac{1}{3}x-2 \end{gathered}[/tex]The correct option is A
Is my answer correct help please
Answers
Answer:
Yes your answer is right !
Step-by-step explanation:
steps
X= 3 and y = 7
So first replace [tex]2^{x}[/tex] with [tex]2^{3}[/tex] an that will give you 8Then 8-Y and so you replace y with 7 and so it becomes
8-7 = 1So the correct answer is D (1)
Hope this helps
~~Wdfads~~
Solve the following addition and subtraction problems.3 km9hm9dam19 m+7km2 dam5sq km95 ha8,994sq m+11sq km11 ha9,010sq m44m−5dm72km47hm2dam−11 km55hm
Answers
As a well accepted rule to solve this problem, we would transform all values to the lower units.
so for the first question:
3 km 9hm 9 dam 19 m + 7 km 2 dam
3,000 m 900 m 90 m 19 m + 7,000 m 20 m
= 4,009 + 7,020
= 11,029 m
The second question:
5 sq.km 95 ha 8,994 sq.m + 11 sq.km 11 ha 9,010 sq.m
5,000,000 sq m 95,0000 sq m 8,994 sq m + 11,000,000 sq m 110,000 sq 9,010 sq m
= 5,103,994 sq m + 11,119,010 sq m
= 16,223,004 sq m
The third question:
44 m - 5 dm
44 m - 0.5 dm
= 43.5 m
The fourth question:
72 km 47 hm 2 dam - 11 km 55 hm
72,000 m 4,700 m 20 m - 11,000 m 5,500 m
= 76,720 m - 16, 500 m
= 60,220 m
Options for this are: 20 of the best selling cameras, same photographer, 100 pictures with each camera, consistent across all cameras 10 point scale, two were from companies who are major advertisers
Answers
It is given that:
A writer for a magazine recently did a test to determine which mid-range digital camera takes the best pictures. Her method is described below.
Which part of the method describes an area of potential bias?
She gathered 20 of the best.selling cameras and used the same photographer to take 100 pictures with each camera .She ensured that the environment and the subject of each picture were consistent across all cameras and used a 10.point scale to determine picture quality. Of the cameras tested, two were from companies who are major advertisers in the magazine.
Now if the reading is done carefully, it can be concluded that the information given by:
"Of the cameras tested, two were from companies who are major advertisers in the magazine." can be considered for a potential bias since the magazine may be pressured by these two companies to give them a higher rating than they deserve.
So the option:two were from companies who are major advertisers is correct.
The function table below is intended to represent the relationship y=-2x-5. However, one of the entries for y does not correctly fit the relationship with x.
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x = 1 , f(x) = -2•1 - 5 = -7
Then it doesnt corresponds to f(1) = 6
Answer is OPTION E)
What is the divisibility rule for 4
A. Last two digits divisible by 4
B. Add all of the digits and divide by 4
C. Last 3 digits divisible by 4
D. Even number
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Answer :- A) Last two digits divisible by 4.
A box contains six red pens, four blue pens, eight green pens, and some black pens. Leslie picks a pen and returns it to the box each time. The outcomes are recorded in the table.a. what is the experimental probability of drawing a green pen?b. if the theoretical probability of drawing a black pen is 1/10, how many black pens are in the box
Answers
given the follwing parameters,
number of times a Red Pen is picked is 8
numbr o f times the Blue Pen is picked is 5
Number of times the Green Pen is picked is 14
Number of times the Black Pen is picked is 3
so,
(a) to get the experimental probability of drawing a Green Pen is,
P = favoured results/all obtained
then,
14/(8+5+14+3)
= 14/30 that is a
(
Two wheelchair ramps, each 10 feet long, lead to the two ends of the entrance porch of Mr. Bell's restaurant. The two ends of the porch are at the same height from the ground, and the start of each ramp is the same distance from the base of the porch. The angle of the first ramp to the ground is 24°.Which statement must be true about the angle of the second ramp to the ground?A. It could have any angle less than or equal to 24°.B. It must have an angle of exactly 24°.C. It could have any angle greater than or equal to 24°.D. Nothing is known about the angle of the second ramp.
Answers
Given statement
The ramps have
- the same height
- the same angle measure relative to the ground
- the two ends of the porch are at the same height from the ground
- the start of each ramp is the same distance from the base of the porch
A pictorial description of the problem is shown below:
Since the two ramps have similar descriptions, the angle measure of the second ramp to the ground would be exactly 24 degrees
Answer: Option B
option b your welcome
Chase and his brother want to improve their personal information for when they startapplying to colleges of their choice. To accomplish this they decide to help the SalvationArmy with delivering hot meals to senior citizens. About a month ago, they decided tokeep track of how many successful deliveries they have each completed. As of today,Chase has successfully delivered 18 out of the 30 meals to senior citizens.Part AHow many more meals would Chase have to deliver in a row in order to have a 75%successful delivery record? Justify your answer.Part BHow many more meals would Chase have to deliver in a row in order to have a 90%successful delivery record? Justify your answer.PartAfter successfully delivering 18 out of 30 meals would Chase ever be able to reach a100% successful delivery record? Explain why or why not.
Answers
Part A.
Chase has successfully delivered 18 out of the 30 meals to senior citizens.
We have to calculate how many more meals (lets call it x) she has to deliver to have a 75% successful delivery record.
In order to do that, (18+x) meals have te be delivered successfully out of (30+x), and the successful meals (18+x) divided by (30+x) has to be 0.75:
[tex]\begin{gathered} \frac{18+x}{30+x}=0.75 \\ 18+x=0.75(30+x) \\ 18+x=22.5+0.75x \\ x-0.75x=22.5-18 \\ 0.25x=4.5 \\ x=\frac{4.5}{0.25} \\ x=18 \end{gathered}[/tex]Chase has to deliver 18 more meals successfully in order to have a 75% success delivery record.
Part B.
We apply the same analysis but we replace 0.75 with 0.9 as the delivery record.
[tex]\begin{gathered} \frac{18+x}{30+x}=0.9 \\ 18+x=0.9(30+x) \\ 18+x=27+0.9x \\ (1-0.9)x=27-18 \\ 0.1x=9 \\ x=\frac{9}{0.1} \\ x=90 \end{gathered}[/tex]Chase has to deliver 90 more meals successfully in order to have a 90% success delivery record.
Part C.
She won't be able to achieve 100% successful delivery record. We can prove it mathematically, but we already know as there are 12 meals that weren't successfully delivered, so we can get close to 100% but it can't never be reached.
Mathematically we have:
[tex]\begin{gathered} \frac{18+x}{30+x}=1 \\ 18+x=30+x \\ x-x=30-18 \\ 0=12 \end{gathered}[/tex]This solution is not valid, so there is no valid solution for x.
Which statement about the graph below is true?
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Answer:
a. The relation is a function because every input has an output.
Step-by-step explanation:
a relation in which for every input there is exactly one output (for every x there is just one y)
quizlet
Answer:
A. The relation is a function because every input has an input
Step-by-step explanation:
A relation is a function as long as there are not multiple outputs for one input. It's okay if there are multiple inputs for one output, like we can see here with points (-6, 1) and (2, 1).
Another way to test if a graphed relation is a function is the vertical line test. Draw vertical lines at multiple spots on the graph and if any of the vertical lines touches 2 points, the graphed relation is not a function.
:)
I need help with some problems on my assignment please help
Answers
The circumcenter of a triangle is the center of a circumference where the three vertex are included. So basically we must find the circumference that passes through points O, V and W. The equation of a circumference of a radius r and a central point (a,b) is:
[tex](x-a)^2+(y-b)^2=r^2[/tex]We have three points which give us three pairs of (x,y) values that we can use to build three equations for a, b and r. Using point O=(6,5) we get:
[tex](6-a)^2+(5-b)^2=r^2[/tex]Using V=(0,13) we get:
[tex](0-a)^2+(13-b)^2=r^2[/tex]And using W=(-3,0) we get:
[tex](-3-a)^2+(0-b)^2=r^2[/tex]So we have a system of three equations and we must find three variables: a, b and r. All equations have r^2 at their right side. This means that we can take the left sides and equalize them. Let's do this with the second and third equation:
[tex]\begin{gathered} (0-a)^2+(13-b)^2=(-3-a)^2+(0-b)^2 \\ a^2+(13-b)^2=(-3-a)^2+b^2 \end{gathered}[/tex]If we develop the squared terms:
[tex]a^2+b^2-26b+169=a^2+6a+9+b^2[/tex]Then we substract a^2 and b^2 from both sides:
[tex]\begin{gathered} a^2+b^2-26b+169-a^2-b^2=a^2+6a+9+b^2-a^2-b^2 \\ -26b+169=6a+9 \end{gathered}[/tex]We substract 9 from both sides:
[tex]\begin{gathered} -26b+169-9=6a+9-9 \\ -26b+160=6a \end{gathered}[/tex]And we divide by 6:
[tex]\begin{gathered} \frac{-26b+160}{6}=\frac{6a}{6} \\ a=-\frac{13}{3}b+\frac{80}{3} \end{gathered}[/tex]Now we can replace a with this expression in the first equation:
[tex]\begin{gathered} (6-a)^2+(5-b)^2=r^2 \\ (6-(-\frac{13}{3}b+\frac{80}{3}))^2+(5-b)^2=r^2 \\ (\frac{13}{3}b-\frac{62}{3})^2+(5-b)^2=r^2 \end{gathered}[/tex]We develop the squares:
[tex]\begin{gathered} (\frac{13}{3}b-\frac{62}{3})^2+(5-b)^2=r^2 \\ \frac{169}{9}b^2-\frac{1612}{9}b+\frac{3844}{9}+b^2-10b+25=r^2 \\ \frac{178}{9}b^2-\frac{1702}{9}b+\frac{4069}{9}=r^2 \end{gathered}[/tex]So this expression is equal to r^2. This means that is equal
A lab assistant needs to create a 1000 ML mixture that is 5% hydroelectric acid. The assistant has solutions of 3.5% and 6% in supply at the lab. Using the variables x and y to represent the number of milliliters of the 3.5% solution and the number of milliliters of the 6% solution respectively, determine a system of equation that describes the situation the situation.Enter the equations below separated by a comma How many milliliters of the 3.5% solution should be used?How many milliliters of 6% solution should be used?
Answers
Given:
A lab assistant needs to create a 1000 ML mixture that is 5% hydroelectric acid.
The assistant has solutions of 3.5% and 6% in supply at the lab.
let the number of milliliters from the solution of 3.5% = x
And the number of milliliters from the solution of 6% = y
so, we can write the following equations:
The first equation, the sum of the two solutions = 1000 ml
So, x + y = 1000
The second equation, the mixture has a concentration of 5%
so, 3.5x + 6y = 5 * 1000
So, the system of equations will be as follows:
[tex]\begin{gathered} x+y=1000\rightarrow(1) \\ 3.5x+6y=5000\rightarrow(2) \end{gathered}[/tex]Now, we will find the solution to the system using the substitution method:
From equation (1)
[tex]x=1000-y\rightarrow(3)[/tex]substitute with (x) from equation (3) into equation (2):
[tex]3.5\cdot(1000-y)+6y=5000[/tex]Solve the equation to find (y):
[tex]\begin{gathered} 3500-3.5y+6y=5000 \\ -3.5y+6y=5000-3500 \\ 2.5y=1500 \\ y=\frac{1500}{2.5}=600 \end{gathered}[/tex]substitute with (y) into equation (3) to find x:
[tex]x=1000-600=400[/tex]So, the answer will be:
Enter the equations below separated by a comma
[tex]x+y=1000,3.5x+6y=5000[/tex]How many milliliters of the 3.5% solution should be used?
400 milliliters
How many milliliters of 6% solution should be used?
600 milliliters
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Answers
By definition, the probability of an event has to be between 0 and 1.
Given that definition the options 1.01, -0.9, -5/6 and 6/5 cannot be the probability of an event.