Answer:
[tex]\displaystyle 24x^{2} - 41x + 12 = 24\left(x - \frac{3}{8}\right) \cdot \left(x - \frac{4}{3}\right) = (8x-3)\cdot (3x - 4)[/tex].
Step-by-step explanation:
Apply the quadratic formula to find all factors. For a quadratic equation in the form
[tex]a\cdot x^{2} + b\cdot x + c = 0[/tex],
where [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] are constants, the two roots will be
[tex]\displaystyle x_1 = \frac{-b + \sqrt{b^{2} - 4\cdot a \cdot c}}{2a}[/tex], and
[tex]\displaystyle x_2 = \frac{-b - \sqrt{b^{2} - 4\cdot a \cdot c}}{2a}[/tex].
For this quadratic polynomial,
[tex]a = 24[/tex],[tex]b = -41[/tex], and[tex]c = 12[/tex].Apply the quadratic formula to find any [tex]x[/tex] value or values that will set this polynomial to zero:
[tex]\displaystyle x_1 = \frac{-(-41) + \sqrt{(-41)^{2} - 4\times 24 \times 12}}{2\times 24} = \frac{3}{8}[/tex].
[tex]\displaystyle x_2 = \frac{-(-41) - \sqrt{(-41)^{2} - 4\times 24 \times 12}}{2\times 24} = \frac{4}{3}[/tex].
Apply the factor theorem to find the two factors of this polynomial:
[tex]\displaystyle \left(x - \frac{3}{8}\right)[/tex] for the root [tex]\displaystyle x = \frac{3}{8}[/tex], and[tex]\displaystyle \left(x - \frac{4}{3}\right)[/tex] for the root [tex]\displaystyle x = \frac{4}{3}[/tex].Keep in mind that simply multiplying the two factors will not reproduce the original polynomial. Doing so assumes that the leading coefficient of [tex]x[/tex] in the original polynomial is one, which isn't the case for this question.
Multiply the product of the two factors by the leading coefficient of [tex]x[/tex] in the original polynomial.
[tex]\displaystyle 24\left(x - \frac{3}{8}\right) \cdot \left(x - \frac{4}{3}\right) = (8x-3)\cdot (3x - 4)[/tex].
Expand to make sure that the factored form is equivalent to the original polynomial:
[tex](8x-3)\cdot (3x - 4)\\ = (8\times 3)x^{2} + ((-3)\times 3 + (-4)\times 8)\cdot x + ((-3)\times (-4))\\ = 24x^{2} - 41x + 12[/tex].
Factor completely 9x2 + 9x -28
Answer:
Your answer is (3x - 4) (3x + 7)
Step-by-step explanation:
Hope my answer has helped you and if not i'm sorry.
Describe the graph of the function.
y =Square root of
x-1 + 4
Answer: Square root of x+3
Step-by-step explanation:
Insert the denominator to 3
ANSWER
The graph of the function starts from (1,4) and moves up right.
EXPLANATION
The given rational function is
[tex]y = \sqrt{x - 1} + 4[/tex]
The base of this function is
[tex]y = \sqrt{x} [/tex]
The -1 under radical sign means the graph of the base function is shifted 1 unit right.
The +4 shows that base function is shifted up by 4 units.
The graph of the function therefore starts from (1,4) and moves up right.
Sal is trying to determine which cell phone and service plan to buy for his mother. The first phone costs $100 and $55 per month for unlimited usage. The second phone costs $150 and $51 per month for unlimited usage. How many months will it take for the second phone to be less expensive than the first phone?
The inequality that will determine the number of months, x, that are required for the second phone to be less expensive is 100 + 55x > 150 + 51x100 + 55x < 150 + 51x100x+ 55 > 150x+ 51100x+ 55 < 150x+ 51.
The solution to the inequality is x > 2.4x < 2.4x < 12.5x > 12.5.
Sal’s mother would have to keep the second cell phone plan for at least 231213 months in order for it to be less expensive
Answer:
a) The first inequality 100+55x>150+51x;
b) The last inequality x>12.5
c) 13 months
Step-by-step explanation:
a) Let x be the number of months.
1. The first phone costs $100 and $55 per month for unlimited usage, then for x months it will cost $55x and in total
$(100+55x)
2. The second phone costs $150 and $51 per month for unlimited usage, then for x months it will cost %51x and in total
$(150+51x)
3. If the second phone must be less expensive than the first phone, then
150+51x<100+55x
or
100+55x>150+51x
b) Solve this inequality:
55x-51x>150-100
4x>50
x>12.5
c) Sal's mother has to keep the second cell phone for at least 13 months (because x>12.5).
Part 1:
The first phone costs $100 and $55 per month for unlimited usage.
Let f(x) be the cost of the first phone and x be the number of months.
Equation forms:
[tex]f(x)=55x+100[/tex]
The second phone costs $150 and $51 per month for unlimited usage.
Let g(x) be the cost of the second phone and x be the number of months.
Equation forms:
[tex]g(x)=51x+150[/tex]
We have to find the inequality that will determine the number of months, x, that are required for the second phone to be less expensive, it is given by:
[tex]g(x)<f(x)[/tex]
[tex]51x+150<55x+100[/tex]
Part 2:
The solution to the inequality is:
[tex]51x+150<55x+100[/tex]
=> [tex]51x-55x<100-150[/tex]
=> [tex]-4x<-50[/tex]
=> [tex]-x<-12.5[/tex]
=> [tex]x>12.5[/tex]
Or rounding off to 13.
Part 3:
Sal’s mother would have to keep the second cell phone plan for at least 13 months in order for it to be less expensive.
There were 3,982 people at the soccer game on Thursday there were 1,886 more people at the soccer game on Saturday how many people in all attended both games
Answer:
9850
Step-by-step explanation:
3982 + 3982 + 1886 = 9850
Answer:
the answer to your question is 5 868
2 cups is to 10 cans as 14 cups is to how many cans
Answer: 80 cans
Step-by-step explanation:
The ratio between cups to cans is 1:5
If we have 14 cups, multiply that by 5 and it is equal to 80 cans
[tex]14[/tex] cups is equals to [tex]70[/tex] number of cans.
What is number ?
" Number is defined as the count of any given quantity."
According to the question,
Given,
[tex]2[/tex] cups [tex]=[/tex] [tex]10[/tex] number of cans
[tex]1[/tex] cup [tex]= \frac{10}{2}[/tex] number of cans
Therefore,
[tex]14[/tex] cups [tex]= \frac{10}{2} \times 14[/tex] number of cans
[tex]= 70[/tex] number of cans
Hence, [tex]14[/tex] cups is equals to [tex]70[/tex] number of cans.
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1.) 2x + y = 3
2.) x - 2y = -1
If equation 1 is multiplied by 2 and then the equations are added, the result is
A.3x = 2
B.3x = 5
C.5x = 5
Answer: C is correct, 5x=5
Step-by-step explanation:
1. When equation 1 is multiplied by 2 it becomes 4x+2y=6
2. Now we have...
4x+2y=6 and x - 2y = -1
Add those together and you have your answer! 5x=5 (simplified is x=1)
What is the surface area of the cube below?
Answer:
96 units squared
Step-by-step explanation:
Answer:
B. 96 Units^2.
Step-by-step explanation:
Got Correct On Assist.
If g(x) = 4(x - 2)2 - 4, complete the following statements.
The axis of symmetry of function f is x =
. The axis of symmetry of function g is x =
Answer:
x = 10
Step-by-step explanation:
Please, use " ^ " to denote exponentiation: g(x) = 4(x - 2)^2 - 4
The vertex is located at (2, -4) which numbers come directly from the '2' in (x - 2) and the "-4" at the end of the equation.
Where is the function f(x) that you mentioned?
The axis of symmetry here is the vertical line that passes through the vertex. Its equation is x = 10.
The axis of symmetry for the function g(x) is x = 2, determined by the standard form of a quadratic equation, which tells us the axis of symmetry is at x = h, where g(x) is in the form a(x - h)^2 + k.
The question asks for the axis of symmetry for function g(x) which is given as g(x) = 4(x - 2)2 - 4. The axis of symmetry for a parabolic function in the form f(x) = a(x - h)2 + k is given by x = h. Thus, in the function g(x), the axis of symmetry is x = 2. If f(x) was provided in a similar quadratic form, its axis of symmetry would similarly be derived from its formula. However, since f(x) was not given in the question, we can't determine its axis of symmetry.
A six sided number cube is rolled twice what is the probability that the first roll is an even number and the second roll is a number greater than 4
The probability that a first roll is an even number and a second role is a number greater than 4 when a six-sided number cube is rolled twice is 1/6.
What is the probability of an event?The probability of an event is the fractional value determining how likely is that event to take place. If the event is denoted by A, the number of outcomes favoring the event A is n and the total number of outcomes is S, then the probability of the event A is given as:
P(A) = n/S.
What are independent events?When the occurrence of one event, doesn't affect the occurrence of the other event, then the two events are independent of each other.
If we have two independent events A and B, then the probability of A and B is given as:
P(A and B) = P(A) * P(B).
How do we solve the given question?We are informed that a six-sided number cube is rolled twice. We are asked what is the probability that a first roll is an even number and a second roll is a number greater than 4.
Let the event of getting an even number on a first roll be A.
∴Number of outcomes favorable to event A (n) = 3 {2, 4, 6}
Total number of outcomes (S) = 6 {1, 2, 3, 4, 5, 6}
∴ The probability of getting an even number on a first roll is:
P(A) = n/S = 3/6 = 1/2.
Let the event of getting a number greater than 4 on a second roll be B.
∴Number of outcomes favorable to event B (n) = 2 {5, 6}
Total number of outcomes (S) = 6 {1, 2, 3, 4, 5, 6}
∴ The probability of getting a number greater than 4 on a second roll is:
P(B) = n/S = 2/6 = 1/3.
∵ Our two events, A and B, are independent of each other, that is, the occurrence of one doesn't affect the occurrence of the other, the probability that a first roll is an even number and a second roll is a number greater than 4 is given by:
P(A and B) = P(A)*P(B) = (1/2)*(1/3) = 1/6.
∴ The probability that a first roll is an even number and a second role is a number greater than 4 when a six-sided number cube is rolled twice is 1/6.
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What is the equation of the line that passes through (2, -7) and (-1, 2)?
Answer:
y = -3x - 1
Step-by-step explanation:
The slope intercept form of the equation of a line is:
y = mx + b
where m is the slope, and b is the y-intercept.
First, we find the slope of the line using the two given points.
m = slope = (y2 - y1)/(x2 - x1) = (2 - (-7))/(-1 - 2) = (2 + 7)/(-3) = 9/(-3) = -3
Now we plug in the slope we found into the equation above.
y = -3x + b
We need to find the value of b, the y-intercept. We use the coordinates of one of the given points for x and y, and we solve for b. Let's use point (2, -7), so x = 2, and y = -7.
y = -3x + b
-7 = -3(2) + b
-7 = -6 + b
Add 6 to both sides.
-1 = b
Now we plug in -1 for b.
y = -3x - 1
ewrite the rational exponent as a radical by extending the properties of integer exponents. 2 to the 3 over 4 power, all over 2 to the 1 over 2 power
[tex]\bf ~\hspace{7em}\textit{rational exponents} \\\\ a^{\frac{ n}{ m}} \implies \sqrt[ m]{a^ n} ~\hspace{10em} a^{-\frac{ n}{ m}} \implies \cfrac{1}{a^{\frac{ n}{ m}}} \implies \cfrac{1}{\sqrt[ m]{a^ n}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{2^{\frac{3}{4}}}{2^{\frac{1}{2}}}\implies 2^{\frac{3}{4}}\cdot 2^{-\frac{1}{2}}\implies 2^{\frac{3}{4}-\frac{1}{2}}\implies 2^{\frac{3-2}{4}}\implies 2^{\frac{1}{4}}\implies \sqrt[4]{2^1}\implies \sqrt[4]{2}[/tex]
3ln(2x) + 7 = 15. how do I do this
Answer:
Step-by-step explanation:
3 ln(2x) + 7 = 15 Subtract 7 from both sides
3 ln(2x) = 15 - 7 Combine
3 ln(2x) = 8 Divide by 3
ln(2x) = 8/3 Do the division. Take the antilog of both sides.
e^ln(2x) = e^(8/3)
2x = 14.392 Divide by 2
x = 14.392
x = 7. 196
Formula One race cars can reach speeds of approximately 100 meters per second. What is this speed in meters per minute?
O 0.6 meters per minute
O 1.5 meters per minute
O 600 meters per minute
O 6,000 meters per minute
Answer:
=6000 m/min
Step-by-step explanation:
There are 60 seconds in 1 minute.
The speed of the race car is 100 m/s. If it covers 100 m in one second, then in one minute it will cover a longer distance calculated as follows.
100 m/s × 60 second/min = 6000 m/min
Answer: Last Option
6,000 meters per minute
Step-by-step explanation:
We know that 60 seconds is equal to one minute.
Then we use this data as a conversion factor.
The speed of the car is 100 meters per second, this is:
[tex]s = 100\ \frac{meters}{seconds}[/tex]
Now we multiply this amount by the conversion factor in the following way:
[tex]100\ \frac{meters}{second} * \frac{60\ second}{1\ minute}= 6000\ \frac{meters}{minute}[/tex]
The answer is:
6,000 meters per minute
find the value of x from the equation
[tex]
\dfrac{8^{3x}}{2^{10}}=\dfrac{4^{2x}}{16} \\
\dfrac{2^{3(3x)}}{2^{10}}=\dfrac{4^{2x}}{4^2} \\
2^{9x-10}=4^{2x-2} \\
2^{9x-10}=2^{2(2x-2)} \\
2^{9x-10}=2^{4x-4} \\
9x-10=4x-4 \\
5x-6=0 \\
5x=6 \\
\boxed{x=\dfrac{6}{5}}
[/tex]
Hope this helps.
r3t40
You have travelled 25 kilometres.
Your friend's map gives the scale as:
20cm:100km.
How many cm on the map have you travelled?
Answer:
5cm
Step-by-step explanation:
okay what you need to know is that:
1cm:5km
5km times 5 is the distance you traveled, 25km
1cm times 5 is 5
Don't get confused with 5km to 5.
It's totally different.
Using the map scale of 20cm to 100km, it is calculated that 25 kilometers in reality is represented by 5 centimeters on the map. This is determined by setting up a proportion between the distance on the map and the actual distance and solving for the unknown value.
To find out how many centimeters on the map represent 25 kilometers of actual distance, we use the map's scale. The scale given is 20cm:100km, which means that for every 20 centimeters on the map, the actual distance represented is 100 kilometers. To solve the problem, we need to set up a proportion.
First, convert the actual distance you want to find on the map from kilometers to centimeters using the scale. So, for every 100 kilometers, we have 20 centimeters on the map. Hence, for 1 kilometer, the distance on the map would be 20 cm divided by 100 km.
20 cm/100 km = X cm/25 km
To find the value of 'X', our unknown value which represents how far you've travelled on the map, we perform cross-multiplication.
100 km * X cm = 20 cm * 25 km
X = (20 cm * 25 km) / 100 km
X = 500 cm / 100
X = 5 cm
So, you have travelled 5 centimeters on the map.
The Green family is a family of six people. They have used 4,885.78 gallons of water so far this month. They cannot exceed 9,750.05 gallons per month during drought season. Write an inequality to show how much water just one member of the family can use for the remainder of the month, assuming each family member uses the same amount of water every month.
Answer:
The inequality is [tex]6x+4,885.78\leq 9,750.05[/tex]
[tex]x\leq 810.71\ gal[/tex]
Step-by-step explanation:
Let
x -----> amount of water that can be used by only one family member for the rest of the month
we know that
[tex]6x+4,885.78\leq 9,750.05[/tex]
Solve for x
Subtract 4,885.78 both sides
[tex]6x\leq 9,750.05-4,885.78[/tex]
[tex]6x\leq 4,864.27[/tex]
Divide by 6 both sides
[tex]x\leq 4,864.27/6[/tex]
[tex]x\leq 810.71\ gal[/tex]
Answer:
Each member can use, x = 810.71 gallons of water for the remaining month.
The equation is 6x + 4885.78 [tex]\leq[/tex] 9750.05
Step-by-step explanation:
Total number of members in Green's family = 6
Let one member uses = x gallons of water
Therefore six member will use = 6x gallons of water
Given the Green's family can use only 9750.05 gallons of water in one month. And they have use so far 4885.78 gallons of water.
Therefore the equation becomes
6x + 4885.78 [tex]\leq[/tex] 9750.05
6x [tex]\leq[/tex] 9750.05 - 4885.78
6x [tex]\leq[/tex] 4864.27
x [tex]\leq[/tex] 4864.27 / 6
x [tex]\leq[/tex] 810.71 gallons
Hence, the equation is 6x + 4885.78 [tex]\leq[/tex] 9750.05
and each member can use 810.71 gallons of water for the remaining days of the month.
Find the distance between the given points.
W(0, 8) and X(0, 12)
Distance =
4
4√(13)
10
Answer:
The distance would be 4 units and you would go 4 to the right
I hope this helps :)
Step-by-step explanation:
Answer: First option
Step-by-step explanation:
The distance between two points is calculated using the following formula
[tex]d=\sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}[/tex]
In this case we have the following points
W(0, 8) and X(0, 12)
Therefore
[tex]x_1 = 0\\x_2 = 0\\y_1=8\\y_2=12[/tex]
[tex]d=\sqrt{(0-0)^2 +(12-8)^2}[/tex]
[tex]d=\sqrt{(12-8)^2}[/tex]
[tex]d=\sqrt{(4)^2}[/tex]
[tex]d=4[/tex]
Drag each description to the correct location on the table. Each description can be used more than once.
Classify each polynomial based on its degree and number of terms.
Answer:
-6 - x^5+3x^2 is cubic, and trinomial
5x^3 - 8x is cubic, and binomial
1/3x^4 is quartic, and monomial
6/7x + 1 is linear, and binomial
-0.7x^2 is quadratic, and monomial
Step-by-step explanation:
Monomial is 1 term
Binomial is 2 terms
Trinomial is 3 terms
- Exponents don't count as terms btw
Answer:
Step-by-step explanation:
find the cube root of 8x7x9x49x3
Answer:
=42
Step-by-step explanation:
The expression 8×7×9×49×3 can be written in its simplest factor form as follows.
8=2³
49=7²
9=3²
Thus the expression becomes:2³×7×3²×7²×3
Combine the indices to the same base.
2³×3³×7³
Finding the cube root involves dividing the index by three.
Thus ∛(2³×3³×7³)= 2×3×7
=42
write an equation in which the quadratic expression 2x^2-2x-13 equals 0
show the expression in factored form and explain what your solutions mean for the equation. SHOW UR WORK
Answer:
x = 3 and -2 are the two solutions.
Step-by-step explanation:
2x^2 -2x -12 = 0
First factor out a 2
2(x^2 - x - 6) = 0
2(x - 3)(x + 2) = 0
2 = 0 x - 3 = 0 x + 2 = 0
2 does not equal zero so this is an extraneous solution
x - 3 = 0 so x = 3
x + 2 = 0 so x = -2
Suppose f(x) = 8(8x – 7) + 8. Solve f(x) = 0 for x
Answer:
x=3/4
Step-by-step explanation:
0=8(8x-7)+8
0=64x-56+8
0=64x-48
48=64x
x=48/64
x=6/8
x=3/4
[tex]8(8x - 7) + 8=0\\64x-56=-8\\64x=48\\x=\dfrac{48}{64}=\dfrac{3}{4}[/tex]
Simplify the expression given below.x+2/4x²+5x+1*4x+1/x²-4
Answer: [tex]\bold{\dfrac{1}{(x+1)(x-2)}}[/tex]
Step-by-step explanation:
[tex]\dfrac{x+2}{4x^2+5x+1}\times \dfrac{4x+1}{x^2-4}\\\\\\\text{Factor the quadratics:}\\\dfrac{x+2}{(4x+1)(x+1)}\times \dfrac{4x+1}{(x-2)(x+2)}\\\\\\\text{Simplify - cross out (4x+1) and (x+2):}\\\dfrac{1}{(x+1)(x-2)}[/tex]
Answer:
[tex]\frac{1}{x^2 - x - 2}[/tex]
Step-by-step explanation:
The given expression is
[tex]\frac{x+2}{4x^2+5x+1}\times \frac{4x+1}{x^2-4}[/tex]
Factorize the denominators.
[tex]\frac{x+2}{4x^2+4x+x+1}\times \frac{4x+1}{x^2-2^2}[/tex]
[tex]\frac{x+2}{4x(x+1)+1(x+1)}\times \frac{4x+1}{(x-2)(x+2)}[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]\frac{x+2}{(x+1)(4x+1)}\times \frac{4x+1}{(x-2)(x+2)}[/tex]
Cancel out common factors.
[tex]\frac{1}{(x+1)}\times \frac{1}{(x-2)}[/tex]
[tex]\frac{1}{(x+1)(x-2)}[/tex]
On further simplification we get
[tex]\frac{1}{x^2 - x - 2}[/tex]
Therefore, the simplified form of the given expression is [tex]\frac{1}{x^2 - x - 2}[/tex].
You have $39 to spend at the music store. Each cassette tape costs $5 and each CD costs $11. Write a linear inequality that represents this situation. Let represent the number of tapes and the number of CDs.
Hello There!
My equation would be 5c + 11d <= 39
C represents cassete tape
D represents disk or the second letter of CD
To solve this I set tapes as X and CDs as Y and then multiplied each variable by the cost, respectively.
Next I set the inequality to be less than or equal to 39, because that is the amount of money you have.
Elm Street is straight. Willard's house is at point H between the school at point S and
the mall at point M.
If SH = 3 miles and HM = 4.5 miles, what is the value of SM in miles?
Answer:
7.5 Miles.
Step-by-step explanation:
3 + 4.5 = 7.5 Miles.
The functions fx) and g(x) are shown on the graph.
f(x) = x2
What is g(x)?
The expression for g(x) is g(x) = x^3, a cubic function that passes through the origin and is slightly narrower than the graph of f(x) = x^2.
Certainly! Let's walk through the step-by-step calculation to determine the expression for g(x) based on the given information.
Given Information:
f(x) = x^2 (quadratic function)
Coordinate marked for g(x) is (2, 8)
g(x) is a parabola slightly narrower than f(x)
The curve g(x) passes through the origin
Expression for g(x):
Since g(x) is a parabola, we consider it as g(x) = ax^2 initially.
The coordinate (2, 8) helps determine the value of a.
g(2) = a(2)^2 = 4a = 8
Solving for a, we get a = 2.
So, g(x) = 2x^2 is the initial expression.
Adjusting for Narrowness:
If g(x) is slightly narrower than f(x), we need to make it narrower than 2x^2.
To achieve this, we can use g(x) = x^3.
Verification:
Check the coordinate (2, 8) in g(x) = x^3:
g(2) = 2^3 = 8
The coordinate (2, 8) satisfies the expression.
what is the value of x? enter your answer in the box.
im pretty sure its 85
please correct me if im wrong!
Answer:
x = 4
Step-by-step explanation:
We require to find RT
Since ΔRST is right use the sine ratio to find RT
sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{2\sqrt{3} }{RT}[/tex]
Multiply both sides by RT
RT × sin60° = 2[tex]\sqrt{3}[/tex] ( divide both sides by sin60° )
RT = [tex]\frac{2\sqrt{3} }{sin60}[/tex] = 4
-----------------------------------------------------------------------
Since ΔRTQ is right use the tangent ratio to find x
tan45° = [tex]\frac{RQ}{RT}[/tex] = [tex]\frac{x}{4}[/tex]
Multiply both sides by 4
4 × tan45° = x, hence
x = 4 × 1 = 4
That is x = 4
the radius of the sphere is 10 units .what is the approximate volume of the sphere use π=3.14
Answer:
V≈4188.79
Step-by-step explanation:
The formula of the volume of a sphere is V=4/3πr^3
Find the soultion(s) to the system of equations. Select all that apply
Answer:
(0,-3)
(3,0)
Step-by-step explanation:
The solutions to the system of equations are where the two graphs cross
The first is at x=0 and y=-3
The second is at x=3 and y=0
Solve the system of equations. 3x+4y+3z=5, 2x+2y+3z=5 and 5x+6y+7z=7
To solve the given system of equations, one can utilize matrix operations, specifically finding the inverse of the coefficient matrix and multiplying it by the constants matrix to solve for the variables x, y, and z.
Solving a System of Equations
To solve the system of linear equations: 3x+4y+3z=5, 2x+2y+3z=5, and 5x+6y+7z=7, we can use methods such as substitution, elimination, or matrix operations. In this case, matrix operations may be more efficient for finding the values of x, y, and z.
Firstly, we must write the system of equations in matrix form (Ax = b), with A being the coefficient matrix, x the variable matrix, and b the constant terms matrix:
A =
| 3 4 3 |
| 2 2 3 |
| 5 6 7 |,
x =
| x |
| y |
| z |,
b =
| 5 |
| 5 |
| 7 |
Next, we find the inverse of matrix A, if it exists, and then multiply it by b to solve for x:
x = A-1 * b
Through matrix operations, we can find A-1. The existence of an inverse is dependent on the determinant of A not being zero. If the determinant is non-zero, the inverse can be used to compute the variables' values. Therefore, we proceed with calculating the determinant and, if possible, the inverse to solve for x, y, and z.
Finally, we multiply the inverse of A (if it exists) by b to get the values for x, y, and z. This involves algebraic steps and matrix multiplication. If any mistake is made, the process requires careful checking and rechecking.
A right cylinder has a radius of 3 and a height of 12. What is its surface area?
Answer:
Surface Area = 283
Step-by-step explanation:
Radius = 3
Height = 12
Formula for suface area is SA= 2πrh+2πr^2
let "r" be for radius
let "h" be for height
SA=2*π*3*12+2*π*3^2
SA=282.74
round to the nearest ones spot and that will make the surface area = 283
The surface area of a right cylinder with a radius of 3 and a height of 12 is calculated using the formula S = 2πr(h + r), resulting in S = 90π or approximately 282.6 square units.
The question is asking to calculate the surface area of a right cylinder with a known radius and height. The formula to find the surface area of a cylinder is S = 2πr(h + r), where r is the radius, and h is the height.
To calculate the surface area of the given cylinder with a radius of 3 and a height of 12, we plug in these values into the formula:
S = 2 × π × 3 × (12 + 3)
This simplifies to S = 2 × π × 3 × 15, which further simplifies to S = 2 × π × 45, and hence S = 90π. By multiplying this by the approximate value of π (3.14), we get S = 282.6 square units as the surface area of the cylinder.
The cross-sectional area (base area) of the cylinder is πr², which is the area of a circle with the same radius as the cylinder. The side surface area, which is the area of the rectangle that would be formed if you 'unrolled' the outer surface of the cylinder, is 2πrh. Adding two times the base area and the side surface area gives you the total surface area of the cylinder.