I'm not sure if the last two apostrophes are part of the quote - "Solve ... " - or if you mean the second derivative [tex]x''[/tex]. I think you mean the first interpretation, but I'll include both cases since they are both solvable.
If the former is correct, separate variables to solve.
[tex]x' = -9tx \implies \dfrac{dx}{dt} = -9tx \implies \dfrac{dx}x = -9t\,dt[/tex]
Integrate both sides to get
[tex]\ln|x| = -\dfrac92 t^2 + C[/tex]
Solve for [tex]x[/tex].
[tex]e^{\ln|x|} = e^{-9/2\,t^2 + C} \implies \boxed{x = Ce^{-9/2\,t^2}}[/tex]
If you meant the latter, then the ODE can be rewritten as
[tex]9t x'' + x' = 0[/tex]
Reduce the order of the equation by substituting [tex]y(t) = x'(t)[/tex] and [tex]y'(t) = x''(t)[/tex].
[tex]9t y' + y = 0[/tex]
Solve for [tex]y'[/tex] and separate variables.
[tex]y' = -\dfrac y{9t} \implies \dfrac{dy}{dt} = -\dfrac y{9t} \implies \dfrac{dy}y = -\dfrac{dt}{9t}[/tex]
Integrate.
[tex]\ln|y| = -\dfrac19 \ln|t| + C[/tex]
Solve for [tex]y[/tex].
[tex]e^{\ln|y|} = e^{-1/9 \,\ln|t| + C} \implies y = Ct^{-1/9}[/tex]
Solve for [tex]x[/tex] by integrating.
[tex]x' = Ct^{-1/9} \implies x = C_1 t^{8/9} + C_2[/tex]
A chemical company makes two brands of antifreeze. The first brand is 20% pure antifreeze, and the second brand is 45% pure antifreeze. In order to obtain 150 gallons of a mixture that contains 35% pure antifreeze, how many gallons of each brand of antifreeze must be used?
The gallons of the first brand of antifreeze should be 60 gallons.
The gallons of the second brand of antifreeze should be 90 gallons.
What are the linear equations that represent the question?0.2f + 0.45s = 52.50 (0.35 x 150) equation 1
f + s = 150 equation 2
Where:
f = gallons of the first antifreezes = gallons of the second antifreezehow many gallons of each brand of antifreeze must be used?
Multiply equation 2 by 0.2
0.2f + 0.2s = 30 equation 3
Subtract equation 3 from equation 2
0.25s = 22.50
s = 22.50 / 0.25
s = 90
f = 150 - 90 = 60 gallons
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What is the value of 11!
Answer:
39916800
Step-by-step explanation:
1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11
A rectangular park had a dirt path across its diagonal that was yards long. The diagonal and the long side of the park formed an angle that measured . A person walked along the sidewalks outside the park, from the start to the end of the path, as shown by the arrows. The diagonal dirt path that cuts across the rectangular park and the sidewalks outside the park form a right triangle. The dirt path is the hypotenuse and it is labeled 100 yards. The start of the sidewalk is adjacent to the 30-degree angle, and it is labeled X. The end of the path is opposite the 30-degree angle, and it is labeled Y. Which expression shows the distance that he walked?
Trigonometry functions are functions that relate two sides of a given triangle with one of its included angles.
The expression that shows the distance that he walked = x + y + 100
= 87 + 50 + 100
= 237 yards
Trigonometry functions are functions that relate two sides of a given triangle with one of its included angles. The required function may be a sine function, a cosine function, or a tangent function.
Such that;
Sin θ = [tex]\frac{Opposite}{Hypotenuse}[/tex]Cos θ = [tex]\frac{Adjacent}{Hypotenuse}[/tex]Tan θ = [tex]\frac{Opposite}{Adjacent}[/tex]The given question can be solved by applying the appropriate trigonometric function.
So that to determine the value of x, we have:
Cos θ = [tex]\frac{Adjacent}{Hypotenuse}[/tex]
Cos 30 = [tex]\frac{x}{100}[/tex]
⇒ x = 100 cos 30
= 100 x 0.866
x = 86.60
Thus, the start of the sidewalk is approximately 87 yards.
To determine the value of sidewalk y, we have:
Sin θ = [tex]\frac{Opposite}{Hypotenuse}[/tex]
Sin 30 = [tex]\frac{y}{100}[/tex]
⇒ y = 100 x Sin 30
= 100 x 0.5
y = 50
Thus the sidewalk which represents the end of the path is 50 yards.
Therefore, the total distance that he walked = 100 + 87 + 50
= 237 yards
The expression that shows the distance that he walked = x + y + 100
= 87 + 50 + 100
= 237 yards
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Ashlee sells lunches for $7.00 and drinks for $2.25. One busy Saturday, Ashlee sold $440.00 worth of lunches and drinks. Which equation represents the relation between the number of lunches sold, x, the number of drinks sold, y, and the amount in dollars that Ashlee sold that day?
Group of answer choices
Answer:
7x + 2.25y = 440
Step-by-step explanation:
Ashlee sells lunches for $7.00 and drinks for $2.25.
One busy Saturday, Ashlee sold $440.00 worth of lunches and drinks.
The number of lunches sold is x.
The number of drinks sold is y.
The appropriate equation will be:
= (7 × x) + (2.25 × y)
7x + 2.25y = 440
A) AB and BC
B) AC and BD
tell if each pair of segments is congruent
a. Line AB and line BC are not congruent because their distance of separation are not equal
b. Line AC and line BD are not congruent because their distance of separation are not equal
How to proof the statementFrom the number line drawn, we have the following deductions;
Line AB = -5 to -2 = 3
Line BC = -2 to 0 = 2
Line AC = -5 to 0 = 5
Line AD = -5 to 5 = 10
Line BD = -2 to 5 = 7
We can see that;
a. Line AB and line BC are not congruent because their distance of separation are not equal
b. Line AC and line BD are not congruent because their distance of separation are not equal
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Will mark brainliest!
Answer:
Area under the graph = 6 sq.units
I've shown the complete calculation over the attached page
A box contains 13 green marbles and 13 white marbles. If the first marble chosen was a white marble, what is the probability of choosing, without replacement, another white marble? Express your answer as a fraction or a decimal number rounded to four decimal places.
[tex]\textbf{Answer:\ \ \large\boxed{\frac{12}{25} }\ or\ 0.4800}[/tex]
Step-by-step explanation:
[tex]\large\boxed{13\ green\ marbles\ and\ 13\ white\ marbles}[/tex]
After a white marble is chosen:
[tex]\large\boxed{13\ green\ marbles\ and\ 12\ white\ marbles}[/tex]
Now there are 25 (13 + 12) total marbles.
Without replacing the white marble we chose already, what is the probability of choosing another white marble?
Well, now there are 12 white marbles to choose from, out of a total of 25 marbles. This means our probability is:
[tex]\large\boxed{\frac{12}{25} }[/tex]
This in decimal form is:
0.4800
A baker uses 3 cups (c) of flour to bake 2 loaves of bread. If 1 pound (lb) of flour is equivalent to 3 c, how many loaves of bread can the baker make with a 10 lb-bag of flour? 3 (Round the answer to the nearest whole number.)
The number of loaves of bread can the baker make with a 10 lb-bag of flour is 20 loaves of bread.
Number of loavesRatio of flour to loaves of bread = 3 cups : 2 loaves
1 pound (lb) of flour = 3 cups of flour10 pounds (lb) of flour = 30 cups of flourNumber of loaves of bread can the baker make with a 10 lb-bag of flour
Equate the ratio of flour to loaves of bread3 : 2 = 30 : x
3/2 = 30/x
3 × x = 2 × 30
3x = 60
x = 60/3
x = 20 loaves of bread
Therefore, the number of loaves of bread can the baker make with a 10 lb-bag of flour is 20 loaves of bread.
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find all the zeros in y=x^2+2x-24
Answer:
x = 4 and x = 6
Step-by-step explanation:
the equation of a circle given the center (–4, 4) and raduis r = 5.
Answer:
x² + y² + 8x - 8y + 7 = 0
Step-by-step explanation:
Formula: (x - a)² + (y - b)²= r²
where,
a = -4
b = 4
Substitute a and b
( x - (-4))² + (y - 4)² = 5²
(x + 4)² + (y - 4)² = 25
Open Bracket( )
(x + 4) (x + 4) = x² + 4x + 4x + 16
(y - 4) (y - 4) = y² - 4y - 4y + 16
x² + 4x + 4x + 16 + y² - 4y - 4y + 16= 25
Collect Like Terms
x² + y² + 8x - 8y + 16 + 16 - 25 = 0
x² + y² + 8x - 8y + 7 = 0
Therefore,
x² + y² + 8x - 8y + 7 = 0
Is The Equation Of The Given Circle.
P = √mx_ £x MX t make x the subject
[tex]p = \sqrt{ \frac{mx}{t} } - t {}^{2}x \: \: make \: x \: the \: subject [/tex]
So, making x subject of the formula, x = [m - 2pt³ ±√(m² - 4pt²m)]/{2t⁵}
How to make x subject of the formula?Since p = √(mx/t) - t²x
So, p + t²x = √(mx/t)
Squaring both sides, we have
(p + t²x)² = [√(mx/t)]²
p² + 2pt²x + t⁴x² = mx/t
Multiplying through by t,we have
(p² + 2pt²x + t⁴x²)t = mx/t × t
p²t + 2pt³x + t⁵x² = mx
p²t + 2pt³x + t⁵x² - mx = 0
t⁵x² + 2pt³x - mx + p²t = 0
t⁵x² + (2pt³ - m)x + p²t = 0
Using the quadratic formula, we find x.
[tex]x = \frac{-b +/-\sqrt{b^{2} - 4ac} }{2a}[/tex]
where
a = t⁵, b = (2pt³ - m) and c = p²tSubstituting the values of the variables into the equation, we have
[tex]x = \frac{-(2pt^{3} - m) +/-\sqrt{(2pt^{3} - m)^{2} - 4(t^{5})(p^{2}t) } }{2t^{5} }\\= \frac{-(2pt^{3} - m) +/-\sqrt{4p^{2} t^{6} - 4pt^{2}m + m^{2} - 4p^{2}t^{6} } }{2t^{5}}\\= \frac{-(2pt^{3} - m) +/-\sqrt{m^{2} - 4pt^{2}m } }{2t^{5}}\\= \frac{m - 2pt^{3} +/-\sqrt{m^{2} - 4pt^{2}m } }{2t^{5}}[/tex]
So, making x subject of the formula, [tex]x = \frac{m - 2pt^{3} +/-\sqrt{m^{2} - 4pt^{2}m } }{2t^{5}}[/tex]
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PLEASE HELP ME .IM IN A HURRY PLEASE HELP ME QUESTION IS DOWN BELOW WORTH 15 POINTS each
Answer: [tex]\frac{10\pi}{3}[/tex] m
Step-by-step explanation:
[tex](2)(\pi)(4) \left(\frac{150}{360} \right)=\frac{10\pi}{3}[/tex]
Lamar, an artist, plans to paint and sell some miniature paintings. He just bought some brushes for $10, and paint and canvas for each painting costs $45; he will sell each painting for $50. Once Lamar sells a certain number of his paintings, he will be breaking even. How many paintings will that be?
Answer:
2
Step-by-step explanation:
If he sells and makes x paintings, his expenses will he 10+45x and his revenue will be 50x.
50x = 45x + 105x = 10x = 2The break-even point is the point at which total cost and total income are equal. For breakeven, Lamar needs to sell 2 paintings.
What is breakeven?In economics, business, and especially cost accounting, the break-even point is the point at which total cost and total income are equal, i.e. "even."
Let the number of paintings that Lamar makes and sells at the break-even point be represented by x.
Given that Lamar bought some brushes for $10, and paint and canvas for each painting cost $45. Therefore, the total cost of making x number of paintings is,
Total cost of x painting = $10 + $45(x)
Also, the selling price of each painting is $50. Therefore, the revenue generated from x paintings is,
Revenue generated = $50(x)
Further, at the breakeven point, the total cost of production of any product is equal to the revenue generated by the product. Therefore, we can write,
Total cost of x painting = Revenue generated
$10 + $45(x) = $50(x)
10 + 45x = 50x
10 = 50x - 45x
10 = 5x
x = 10/5
x = 2
Hence, For breakeven Lamar needs to sell 2 paintings.
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A shipping company charged $15 to ship a package and 5% to insure the package.What was the total cost to ship this particular item?
Answer:
5% of 15 is 0.75 i think
Step-by-step explanation:
15/5% is 0.75
Evaluate 3|12−x|−4 when x=15
Answer:
5
Step-by-step explanation:
3 x |12 - x| - 4
3 x |12 - 15| - 4
3 x 3 - 4
9 - 4
5
IM IN A HURRY PLEASE HELP ME QUESTION IS DOWN BELOW WORTH 15 POINTS each
In the given figure, the measure of the central angle CAD is 80°, the major arc is arc CBD, and minor arc is arc CD. The measure of arc BEC is 2.27r and that of arc BC is 0.87r.
About the Central Angle:
An angle formed by two radii of a circle is known as a central angle. Thus, arc BC and arc CD both subtends central angles at the center.
Since BD is the diameter of the circle,
∠BAC + CAD = 180°
It is given that ∠BAC = 100°
⇒ ∠CAD = 180° - 100°
⇒ ∠CAD = 80°
About Major Arc:
The arc which subtends an angle greater than 180° at the center, is called a major arc.
Angle subtended by arc BEC = 360° - m(arc CD)
= 360° - 80°
= 280° > 180°
∴ Arc BEC is the major arc
About Minor Arc:
The arc which subtends an angle less than 180° at the center, is called a minor arc.
⇒ Arc CD is the minor arc.
Calculating arc BEC and arc BC:
Let us assume the radius of the circle is r.
Then, the formula of the measure of an arc is given by,
θ × (π/180) × r
Here, θ is the angle ( in degrees) subtended by the arc at the center.
Arc BEC = 260 × (π/180)r ......... [Put π = 3.14]
= 2.27r
Similarly, arc BC = 100 ×(π/180) × r .......... [Put π = 3.14]
= 0.87r
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2. What is the solution for the system of equations?
16x - 32y = 27
8x - 16 = 16y
a) Use the linear combination (elimination) method to solve the system of equations.
b) What does the solution tell you about the two lines of the system?
Answer:
no solution
Step-by-step explanation:
16x - 32y = 27 → (1)
8x - 16 = 16y ( subtract 16y from both sides )
8x - 16 - 16y = 0 ( add 16 to both sides )
8x - 16y = 16 → (2)
multiply (2) by - 2 and add to (1)
- 16x + 32y = - 32 → (3)
add (1) and (3) term by term
0 + 0 = - 5
0 = - 5 ← not possible
this indicates the system has no solution
(b)
the solution to a system is the point of intersection of the 2 lines
since there is no solution, no point of intersection, then
this indicates the lines are parallel and never intersect
Answer:
a) no solution
b) the two lines never intersect
Step-by-step explanation:
Given system of equations:
[tex]\begin{cases} 16x-32y=27\\8x-16=16y \end{cases}[/tex]
Part (a)To solve by linear combination (elimination):
Step 1
Write both equations in standard form: Ax + By = C
[tex]\implies 16x-32y=27[/tex]
[tex]\implies 8x-16y=16[/tex]
Step 2
Multiply one (or both) of the equations by a suitable number so that both equations have the same coefficient for one of the variables:
[tex]\implies 16x-32y=27[/tex]
[tex]\implies 2(8x-16y=16) \implies 16x-32y=32[/tex]
Step 3
Subtract one of the equations from the other to eliminate one of the variables:
[tex]\begin{array}{l r l}& 16x-32y & = 32\\- & 16x-32y & = 27\\\cline{1-3}& 0 & =\:\: 5\end{array}[/tex]
Therefore, as 0 ≠ 5, there is no solution to this system of equations.
Part (b)
The solution to a system of equations is the point(s) of intersection.
As there is no solution to the given system of equations, this tells us that the two lines never intersect.
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The two-way frequency table below shows the preferred communication method of employees at a company, based on years of employment with the company.
Text
Message Instant
Message Phone Call Email Total
0 to 7 years 36 49 8 21 114
8 or more years 12 22 19 43 96
Total 48 71 27 64 210
What percentage of employees with 8 or more years at the company reported that email is their preferred method of communication?
A.
48.84%
B.
20.48%
C.
67.19%
D.
44.79%
Using it's concept, the percentage of employees with 8 or more years at the company reported that email is their preferred method of communication is:
D. 44.79%.
What is a percentage?The percentage of an amount a over a total amount b is given by a multiplied by 100% and divided by b, that is:
P = a/b x 100%
In this problem, there are 96 employees with 8 or more years of experience, of which 43 prefer email, hence the percentage is:
P = 43/96 x 100% = 44.79%.
Hence option D is correct.
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HELPPPP PLSSSSSSSSS!!!!!!!!??!
Answer:
(3+3) x (3+1)
Step-by-step explanation:
nice
We have give 4 numbers that are 1,3,3,3. we have to apply operations on it to make it 24.
Solution :» (1 + 3) × (3 + 3)
» (4) × (6)
» 24
Here's our answer..!!
Given angle YMC = 170 degrees, find the measure of angle AMC.
Answer:
i think 10 because in triangle there is 180 and 180-170 = 10
Answer:
Step-by-step explanation:
hjhj
A teacher records the number of students present in her 1st period class each day. This count is a ___________ random variable.
A. Discrete
B. Countable
C. Continuous
D. Finite
This count is a discrete random variable (option A).
What is a discrete random variable?A discrete random variable is a variable that contains integers that can only be a limited number of possible values. A discrete random variable is can contain only a finite set of numbers .
An example of discrete random variable is the number of students in the first period class. It is impossible for the number of students in the class to go on indefinitely.
Discrete random variable has the following properties:
It is finiteIt is numericIt is countableIt contains non-negative integers.A continous random variable is a variable that has an infinite number.
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Questions are in the pictures
The values of h and r to maximize the volume are r = 4 and h = 2
The formula for h in terms of rFrom the question, we have the following equation
2r + 2h = 12
Divide through by 2
r + h = 6
Subtract r from both sides of the equation
h = 6 - r
Hence, the formula for h in terms of r is h = 6 - r
Formulate a function V(r)The volume of a cylinder is
V = πr²h
Substitute h = 6 - r in the above equation
V = πr²(6 - r)
Hence, the function V(r) is V = πr²(6 - r)
The single critical pointV = πr²(6 - r)
Expand
V = 6πr² - πr³
Integrate
V' = 12πr - 3πr²
Set to 0
12πr - 3πr² = 0
Divide through by 3π
4r - r² = 0
Factor out r
r(4 - r) = 0
Divide through by 4
4 - r = 0
Solve for r
r = 4
Hence, the single critical point on the interval [0. 6] is r = 4
Prove that the critical point is a global maximumWe have:
V = πr²(6 - r)
and
V' = 12πr - 3πr²
Determine the second derivative
V'' = 12π - 6πr
Set r = 4
V'' = 12π - 6π* 4
Evaluate the product
V'' = 12π - 24π
Evaluate the difference
V'' = -12π
Because V'' is negative, then the single critical point is a global maximum
The values of h and r to maximize the volumeWe have
r = 4 and h = 6 - r
Substitute r = 4 in h = 6 - r
h = 6 - 4
Evaluate
h = 2
Hence, the values of h and r to maximize the volume are r = 4 and h = 2
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Solve the equation: (4x - 5)^4 = 81.
hope this helps
by
aman10we
The value of x after solving this equation is 2
What is a polynomial?Polynomial is made up of two terms, namely Poly (meaning “many”) and Nominal (meaning “terms.”). A polynomial is an expression composed of variables, constants, and exponents, combined using mathematical operations such as addition, subtraction, multiplication, and division (No division operation by a variable). Based on the number of terms present in the expression, it is classified as monomial, binomial, and trinomial. For example P(x) = x2-5x+11
Given here, the equation as : (4x - 5)^4 = 81.
(4x - 5)^4 = 3⁴.
4x - 5 = 3
x = 2
Hence, the value of x is equal to 2
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An unknown radioactive element decays into non-radioactive substances. In 340
days the radioactivity of a sample decreases by 61 percent.
(a) What is the half-life of the element?
half-life: ? (days)
(b) How long will it take for a sample of 100 mg to decay to 81 mg?
time needed: ? (days)
(a) The half-life of the unknown radioactive element is 250.28 days.
(b) The time taken for a sample of 100 mg to decay to 81 mg is 76.1 days.
Half life of the unknown radioactive elementN(t) = N₀(0.5)^t/h
where;
t is time of decayh is half lifeN₀ is initial massN(t) remaining mass at time, tin 340 days; N(340) = N₀(0.39);
1 - 0.61 = 0.39
N(340) = N₀(0.5)^340/h
N(340)/N₀ = 0.5^340/h
0.39 = 0.5^340/h
log(0.39) = 340/h x log(0.5)
log(0.39) /log(0.5) = 340/h
1.358 = 340/h
h = 340/1.358
h = 250.28 days
Time taken for the sample to decay 81 mg81 = 100(0.5)^t/250.28
81/100 = (0.5)^t/250.28
0.81 = (0.5)^t/250.28
log(0.81) = t/250.28 x log(0.5)
log(0.81) / log(0.5) = t/250.28
0.304 = t/250.28
0.304(250.28) = t
76.1 days = t
Thus, the half-life of the unknown radioactive element is 250.28 days and the time taken for a sample of 100 mg to decay to 81 mg is 76.1 days.
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Allison went to the grocery store and bought bottles of soda and bottles of juice. Each bottle of soda has 45 grams of sugar and each bottle of juice has 20 grams of sugar. Allison purchased a total of 11 bottles of juice and soda which collectively contain 445 grams of sugar. Write a system of equations that could be used to determine the number of bottles of soda purchased and the number of bottles of juice purchased. Define the variables that you use to write the system.
Using a system of equations, it is found that Allison bought 2 bottles of juice and 9 bottles of soda.
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
For this problem, the variables are:
Variable x: bottles of juice purchased.Variable y: bottles of soda purchased.Allison purchased a total of 11 bottles of juice, hence:
x + y = 11 -> x = 11 - y.
These 11 bottles contain 445 grams of sugar, hence, considering the amounts of each bottle, we have that:
20x + 45y = 445
Since x = 11 - y:
20(11 - y) + 45y = 445
25y = 225
y = 225/25
y = 9.
x = 11 - y = 11 - 9 = 2.
She bought 2 bottles of juice and 9 bottles of soda.
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Use cos a cos b=1/2 [cos (a + b) + cos (a-b)]to derive cos x + cos y= cos 2 (x+y/2) cos (x-y/2) .
The trigonometric identity cos x + cos y = 2 cos (x + y/2) cos (x - y/2), is derived using the trigonometric identity cos a cos b=1/2 [cos (a + b) + cos (a-b)].
In the question, we are asked to derive the trigonometric identity, cos x + cos y = 2 cos (x + y/2) cos (x - y/2), using the trigonometric identity cos a cos b=1/2 [cos (a + b) + cos (a-b)].
We are given the trigonometric identity cos a cos b=1/2 [cos (a + b) + cos (a-b)].
Substituting a = x + y/2 and b = x - y/2 in this, we get:
cos (x + y/2) cos (x - y/2) = 1/2[cos (x + y/2 + x - y/2) + cos (x + y/2 - x - y/2)],
or, cos (x + y/2) cos (x - y/2) = 1/2[ cos (2x/2) + cos (2y/2) ],
or, 2 cos (x + y/2) cos (x - y/2) = cos x + cos y, which on inter-changing the sides, gives us:
cos x + cos y = 2 cos (x + y/2) cos (x - y/2), which is the required trigonometric identity.
Thus, the trigonometric identity cos x + cos y = 2 cos (x + y/2) cos (x - y/2), is derived using the trigonometric identity cos a cos b=1/2 [cos (a + b) + cos (a-b)].
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The provided question is incorrect. The correct question is:
"Use cos a cos b=1/2 [cos (a + b) + cos (a-b)]to derive cos x + cos y = 2 cos (x + y/2) cos (x - y/2) ."
IM IN A HURRY PLEASE HELP ME QUESTION IS DOWN BELOW WORTH 15 POINTS each
Answer:
22.34021442
Step-by-step explanation:
22.34021442
The coefficient of 2(3)(6)Q is
The coefficient of 2(3)(6)Q is 2(3)(6)
How to determine the coefficient?The expression is given as:
2(3)(6)Q
For an expression
AQ
Where A is a number or product of numbers and Q is a variable
The number A represents the coefficient
By comparing:
AQ and 2(3)(6)Q
We have
A = 2(3)(6)
Hence, the coefficient of 2(3)(6)Q is 2(3)(6)
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Please help me with the first question!
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
[tex]\qquad❖ \: \sf \: \angle C = 63 \degree[/tex]
[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
[tex]\qquad❖ \: \sf \:x + 4x + 12 + x=180°[/tex]
( Angle EBA= Angle DBC, and the three angles sum upto 180° due to linear pair property )
[tex]\qquad❖ \: \sf \:6x + 12 = 180[/tex]
[tex]\qquad❖ \: \sf \:6x = 180 - 12[/tex]
[tex]\qquad❖ \: \sf \:6x = 168[/tex]
[tex]\qquad❖ \: \sf \:x = 28 \degree[/tex]
Next,
[tex]\qquad❖ \: \sf \: \angle C + \angle D + x = 180°[/tex]
[tex]\qquad❖ \: \sf \: \angle C +3x + 5 + x = 180°[/tex]
[tex]\qquad❖ \: \sf \: \angle C +4x = 180 - 5[/tex]
[tex]\qquad❖ \: \sf \: \angle C +4(28) = 175[/tex]
( x = 28° )
[tex]\qquad❖ \: \sf \: \angle C +112 = 175[/tex]
[tex]\qquad❖ \: \sf \: \angle C = 175 - 112[/tex]
[tex]\qquad❖ \: \sf \: \angle C = 63 \degree[/tex]
[tex] \qquad \large \sf {Conclusion} : [/tex]
[tex]\qquad❖ \: \sf \: \angle C = 63 \degree[/tex]
A chemist has three different acid solutions. The first acid solution contains
20
%
acid, the second contains
30
%
and the third contains
60
%
. They want to use all three solutions to obtain a mixture of
72
liters containing
35
%
acid, using
2
times as much of the
60
%
solution as the
30
%
solution. How many liters of each solution should be used?
Let [tex]x,y,z[/tex] denote the amounts (in liters) of the 20%, 30%, and 60% solutions used in the mixture, respectively.
The chemist wants to end up with 72 L of solution, so
[tex]x+y+z=72[/tex]
while using twice as much of the 60% solution as the 30% solution, so
[tex]z = 2y[/tex]
The mixture needs to have a concentration of 35%, so that it contains 0.35•75 = 26.25 L of pure acid. For each liter of acid solution with concentration [tex]c\%[/tex], there is a contribution of [tex]\frac c{100}[/tex] liters of pure acid. This means
[tex]0.20x + 0.30y + 0.60z = 26.25[/tex]
Substitute [tex]z=2y[/tex] into the total volume and acid volume equations.
[tex]\begin{cases}x+3y = 72 \\ 0.20x + 1.50y = 26.25\end{cases}[/tex]
Solve for [tex]x[/tex] and [tex]y[/tex]. Multiply both sides of the second equation by 5 to get
[tex]\begin{cases}x+3y = 72 \\ x + 7.50y = 131.25\end{cases}[/tex]
By elimination,
[tex](x+3y) - (x+7.50y) = 72 - 131.25 \implies -4.50y = -59.25 \implies \boxed{y=\dfrac{79}6} \approx 13.17[/tex]
so that
[tex]x+3\cdot\dfrac{79}6 = 72 \implies x = \boxed{\dfrac{65}2} = 32.5[/tex]
and
[tex]z=2\cdot\dfrac{79}6 = \boxed{\dfrac{79}3} \approx 26.33[/tex]
