Let A, B, and W denote the sets of students apply to Addis Ababa Uni (A), Bahir Dar Uni (B), or Wachemo Uni (W). Let U denote the universal set of all students in the class.
We're given the cardinalities of several sets:
• total number of students: [tex]n(U) = 60[/tex]
• A applicants: [tex]n(A) = 30[/tex]
• B applicants: [tex]n(B) = 25[/tex]
• W applicants: [tex]n(W) = 24[/tex]
• A and B applicants: [tex]n(A\cap B) = 11[/tex]
• A and W applicants: [tex]n(A \cap W) = 6[/tex]
• B and W applicants: [tex]n(B\cap W) = 9[/tex]
• non-applicants: [tex]n(U \setminus (A\cup B\cup W)) = 4[/tex]
The last cardinality tells us [tex]n(A\cup B\cup W) = 60-4 = 56[/tex] students applied anywhere at all.
We want to find [tex]n(A\cap B\cap W)[/tex], the number of students that applied to each of the three universities.
By the inclusion/exclusion principle,
[tex]n(A\cup B\cup W) = n(A) + n(B) + n(W) \\\\~~~~~~~~~~~~~~~~~~~~~~~~ - n(A\cap B) - n(A\cap W) - n(B\cap W)\\ \\~~~~~~~~~~~~~~~~~~~~~~~~ + n(A\cap B\cap W)[/tex]
That is, we count up all the students in the sets A, B, and W, then subtract the number of students in each pairwise intersection to not double-count, then add back the number of students in the intersection of all three sets since it was removed in the previous step.
Now solve.
[tex]56 = 30 + 25 + 24 - 11 - 6 - 9 + n(A\cap B\cap W)[/tex]
[tex]\implies n(A\cap B\cap W) = \boxed{3}[/tex]
If (x, 68, 85) form a Pythagorean Triple, what is the value of x?
Answer:
x = 51Step-by-step explanation:
If (x, 68, 85) form a Pythagorean Triple, what is the value of x?
to solve we again use Pythagoras, x is the smallest number, so 68 is a cathetus and 85 the hypotenuse
x = √(85²- 68²)
x = (7225 - 4624)
x = √2601
x = 51
Answer:
x = 51
Step-by-step explanation:
The missing value of the Pythagorean triple can be found using the Pythagorean theorem, or it can be found by comparing the values in the triple to known triples.
What are Pythagorean triples?A Pythagorean triple is a set of integers {a, b, c} that satisfy the equation of the Pythagorean theorem:
a² +b² = c²
The smallest such triple is {3, 4, 5}. It is also the only triple that is an arithmetic sequence. Other triples of small integers are ...
{5, 12, 13}, {7, 24, 25}, {8, 15, 17}
There are an infinite number of "primitive" Pythagorean triples, ones that are not multiples of another triple.
What is this triple?The given values of the triple have the ratio ...
68/85 = (4·17)/(5·17) = 4/5
Only the values in the {3, 4, 5} triple and its multiples will have this ratio.
The value of x is 3·17 = 51, so the triple is {51, 68, 85}.
x = 51
__
Additional comments
Any primitive triple will have two odd numbers.
The ratios of numbers in a primitive triple are unique to that triple. That is, the numbers are mutually prime.
For any pair of positive integers m > n, there is a Pythagorean triple {2mn, m²-n², m²+n²}. Such triples will not be primitive if m and n have the same parity.
__
The above triple can be verified using the Pythagorean theorem:
51² +68² = 2601 +4624 = 7225 = 85²
find p + q
someone pls help
Answer:
70
Step-by-step explanation:
45+90=135
180-45=45
q=45
45+65=110
180-110=70
70-45=25
p=25
45+25=70
Answer: 70°
Step-by-step explanation:
We know that the sum of all 3 angles in a triangle is 180°. Hence, we can just set all measures to be equal to 180° and solve for the variables.
Left Triangle[tex]p+65+90=180\\p+155=180\\p=25[/tex]
Right Triangle[tex]45+90+q=180\\135+q=180\\q=45[/tex]
Sum[tex]p+q=25+45=70[/tex]
The sum of p and q is 70°.
a In a concert band, the probability that a member is in the brass section is
0.50. The probability that a member plays trombone, given that he or she is in
the brass section, is 0.24.
What is the probability that a randomly selected band member is in the brass
section and plays trombone?
A. 0.26
B. 0.74
C. 0.12
D. 0.48
Answer:
C. 0.12
Step-by-step explanation:
Let's use a sample number of 100
Out of 100 members, 50 are in the brass section
Out of those 50, 24% play the trombone. This means that 12 people total play the trombone.
12/100 is .12.. That is the final answer
find the equation of the tangent line at the given point on the following curve.
x^2+y^2=8, (-2,-2)
Using derivatives, the equation of the tangent line is: y + 2= -(x + 2)
What is the equation to the tangent line of a function f(x) at point (x0, y0)?The equation is:
[tex]y - y_0 = m(x - x_0)[/tex]
In which m is the derivative at point [tex](x_0, y_0)[/tex].
The function is:
[tex]x^2 + y^2 = 8[/tex].
Applying implicit differentiation, the derivative is:
[tex]2x\frac{dx}{dx} + 2y\frac{dy}{dx} = 0[/tex]
[tex]2ym = -2x[/tex]
[tex]m = -\frac{x}{y}[/tex]
We have that x = y = -2, hence m = -1 and the equation is:
y + 2= -(x + 2)
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Please answer the questions below.
The volumes of the right prisms are listed below:
1 / 9 ft³39 / 512 in³7 / 384 cm³1 / 12 cm³3 / 25 m³3 / 8 yd³What is the volume of the prisms?In this problem we have six cases of right prisms with rectangular bases, the volume of right prisms (V), in cubic length units, is equal to the product of the area of the base (A), in square length units, and the height (h), in length units. The equation of the volume of the right prism is shown below:
V = w · l · h (1)
Where:
w - Width of the base, in length units.l - Length of the base, in length units.h - Height of the prism, in length units.Now we proceed to calculate each volume:
Prism 1 (w = 2 / 3 ft, l = 1 / 3 ft, h = 1 / 2 ft)
V = (2 / 3 ft) · (1 / 3 ft) · (1 / 2 ft)
V = 1 / 9 ft³
Prism 2 (w = 1 / 8 in, l = 3 / 4 in, h = 13 / 16 in)
V = (1 / 8 in) · (3 / 4 in) · (13 / 16 in)
V = 39 / 512 in ³
Prism 3 (w = 7 / 8 cm, l = 1 / 4 cm, h = 1 / 12 cm)
V = (7 / 8 cm) · (1 / 4 cm) · (1 / 12 cm)
V = 7 / 384 cm³
Prism 4 (w = 1 / 7 cm, l = 7 / 8 cm, h = 2 / 3 cm)
V = (1 / 7 cm) · (7 / 8 cm) · (2 / 3 cm)
V = 1 / 12 cm³
Prism 5 (w = 9 / 10 m, l = 2 / 3 m, h = 1 / 5 m)
V = (9 / 10 m) · (2 / 3 m) · (1 / 5 m)
V = 3 / 25 m³
Prism 6 (w = 7 / 8 yd, l = 6 / 7 yd, h = 1 / 2 yd)
V = (7 / 8 yd) · (6 / 7 yd) · (1 / 2 yd)
V = 3 / 8 yd³
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A recent tax bill proposed raising the income-tax rate on families earning at least $200,000 a year by 25%, to 25%. What was the previous tax rate ?
The previous tax rate was 20%
What is tax rate?
Tax rate is the percentage of the earnings tax payers would pay to the government from their earnings as taxes.
It should be borne in mind that the new tax rate is 25% which has increased by 25% compared to the previous tax rate, which is the unknown.
The mathematical relationship between the current and prior tax rates is as shown below:
Current tax rate=previous tax rate*(1+% increase in tax rate)
Current tax rate=25%
previous tax rate=X
% increase in tax rate=25%
25%=X*(1+25%)
25%=X*1.25
X=25%/1.25
X=previous tax rate=20%
Technically speaking, an increase of 25% in 20% is 5%(i.e. 20%*25%=5%), which when added to the original 20% would give a tax rate of 25%.That the shortcut to affirming that the new tax rate of 25% is correct
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i need help with number 70
The volume of the rectangular prism is of 28.6875 mm³.
What is the volume of a rectangular prism?
The volume of rectangular prism of length l, width w and height h is given by:
V = lwh.
For this problem, the dimensions are:
2(1/8)mm = 2 + 1/8 = 2.125 mm.6(3/4)mm = 6 + 3/4 = 6.75 mm.2 mm.Hence the volume is:
V = 2.125 x 6.75 x 2 = 28.6875 mm³.
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Polygon ABCD is rotated 90º counterclockwise about the origin to create polygon A′B′C′D′. Match each set of coordinates to the vertices of polygon A′B′C′D′. (-1, 3)
B′
(-2, 2)
C′
(-2, 1)
D′
(-1, 1)
Answer:Rules for rotating points about the origin
1. The point (a,b) rotated 90° counterclockwise about the origin
becomes (-b,a).
A′ (-1, 3) B′ (-2, 2) C′ (-2, 1) D′ (-1, 1)
A (a,b)-----------------A′ (-b,a)=(-1, 3)----------->A( 3,1)
B (a,b)-----------------B′ (-b,a)=(-2, 2)----------->B( 2,2)
C (a,b)-----------------C′ (-b,a)=(-2, 1)----------->C( 1,2)
D (a,b)-----------------D′ (-b,a)=(-1, 1)----------->D( 1,1)
Step-by-step explanation:
The graph of f(x) = x3 + x2 – 9x – 9 is shown.
Based on the graph, what are the solutions of x3 + x2 – 9x – 9 = 0?
x = –1, 3
x = –3, –1
x = –3, –1, 3
x = –9, –3, –1, 3
Answer:
the third option, x = –3, –1, 3
Step-by-step explanation:
Which is the statement is true about f(x)=-2/3|x+4|-6
Answer: The graph of f(x) is horizontally stretched
Question
The epicenter of an earthquake is the point on Earth's surface directly above the earthquake's origin. A seismograph can be used to determine the distance to the epicenter of an earthquake. Seismographs are needed in three different places to locate an earthquake's epicenter. Find the location of the earthquake's epicenter if it is 2 miles away from A(−2,5), 4 miles away from B(2,3), and 7 miles away from C(−2,−4).
Answer:
The epicenter is ( - 2, 3)Step-by-step explanation:
Let the epicenter be E(x, y).
Use the distance formula for each given distance:
AE² = (x + 2)² + (y - 5)² = 2²BE² = (x - 2)² + (y - 3)² = 4²CE² = (x + 2)² + (y + 4)² = 7²Use the first and third equations, considering both have same term for x, and solve for y:
(y - 5)² - 4 = (y + 4)² - 49y² - 10y + 25 - 4 = y² + 8y + 16 - 4918y = 54y = 3Substitute y into one of equations and solve for x:
(x + 2)² + (3 - 5)² = 4(x + 2)² + 4 = 4(x + 2)² = 0x + 2 = 0x = - 2The epicenter is E( - 2, 3)
Answer:
(-2, 3)
Step-by-step explanation:
Given:
A = (-2, 5) → 2 miles awayB = (2, 3) → 4 miles awayC = (-2, -4) → 7 miles awayModel each given point as the center of a circle and the distance the epicenter is away from the point as the circle's radius.
The epicenter's location will be the point of intersection of the three circles.
Equation of a circle
[tex](x-a)^2+(y-b)^2=r^2[/tex]
where (a, b) is the center and r is the radius
Circle A
center = (-2, 5)radius = 2 miles[tex]\implies (x+2)^2+(y-5)^2=4[/tex]
[tex]\implies x^2+4x+4+y^2-10y+25=4[/tex]
[tex]\implies x^2+4x+y^2-10y+25=0[/tex]
Circle B
center = (2, 3)radius = 4 miles[tex]\implies (x-2)^2+(y-3)^2=16[/tex]
[tex]\implies x^2-4x+4+y^2-6y+9=16[/tex]
[tex]\implies x^2-4x+y^2-6y-3=0[/tex]
Circle C
center = (-2, -4)radius = 7 miles[tex]\implies (x+2)^2+(y+4)^2=49[/tex]
[tex]\implies x^2+4x+4+y^2+8y+16=49[/tex]
[tex]\implies x^2+4x+y^2+8y-29=0[/tex]
To find the point of intersection of the three circles, solve simultaneously.
Substitute equation B into equation A to eliminate x² and y²:
[tex]\implies x^2-4x+y^2-6y-3=x^2+4x+y^2-10y+25[/tex]
[tex]\implies -4x-6y-3=4x-10y+25[/tex]
Rearrange to isolate y:
[tex]\implies -6y-3=8x-10y+25[/tex]
[tex]\implies 4y-3=8x+25[/tex]
[tex]\implies 4y=8x+28[/tex]
[tex]\implies y=2x+7[/tex]
Substitute the expression for y into equation C and simplify:
[tex]\implies x^2+4x+(2x+7)^2+8(2x+7)-29=0[/tex]
[tex]\implies x^2+4x+4x^2+28x+49+16x+56-29=0[/tex]
[tex]\implies 5x^2+48x+76=0[/tex]
Substitute the expression for y into equation B and simplify:
[tex]\implies x^2-4x+(2x+7)^2-6(2x+7)-3=0[/tex]
[tex]\implies x^2-4x+4x^2+28x+49-12x-42-3=0[/tex]
[tex]\implies 5x^2+12x+4=0[/tex]
Equate the equations to eliminate 5x² and solve for x:
[tex]\implies 5x^2+48x+76=5x^2+12x+4[/tex]
[tex]\implies 48x+76=12x+4[/tex]
[tex]\implies 36x+76=4[/tex]
[tex]\implies 36x=-72[/tex]
[tex]\implies x=-2[/tex]
Substitute the found value of x into the found expression for y:
[tex]\implies y=2(-2)+7[/tex]
[tex]\implies y=-4+7[/tex]
[tex]\implies y=3[/tex]
Therefore, the point of intersection of the three circles if (-2, 3) and hence the location of the earthquake's epicenter is (-2, 3).
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If the function is:
What is f(-4)?
Answer:
D. -11
Step-by-step explanation:
Piecewise functions have multiple pieces of curves/lines where each piece corresponds to its definition over an interval.
Given piecewise function:
[tex]f(x)=\begin{cases}2x-3 & \textsf{if }x \leq -4\\x^2 & \textsf{if }-4 < x < 0\\x+5 & \textsf{if }x \geq 0\end{cases}[/tex]
Therefore, the function has three definitions:
[tex]f(x)=2x-3 \quad \textsf{ when x is equal to or less than -4}[/tex]
[tex]f(x)=x^2 \quad \textsf{ when x is greater than -4 or less than zero}[/tex]
[tex]f(x)=x+5 \quad \textsf{ when x is greater than zero}[/tex]
f(-4) means to substitute x = -4 into the function.
As x = -4 satisfies the interval of the first piece of the function, substitute x = -4 into f(x) = 2x - 3:
[tex]\begin{aligned}\implies f(-4) & =2(-4)-3\\& = -8 -3\\& = -11\end{aligned}[/tex]
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Answer is D. -11
Answer:
Solution Given:
f(-4) depends on the condition x≤-4,
so
f(-4)=2*-4-3=-8-3=-11
Which logarithmic equation is equivalent to 82 = 64?
2 = log Subscript 6 Baseline 64
2 = log Subscript 64 Baseline 8
8 = log Subscript 2 Baseline 64
64 = log Subscript 2 Baseline 8
Answer: Choice B
[tex]2 = \log_{8}(64)[/tex]
======================================================
Explanation:
The original equation is [tex]8^2 = 64[/tex] with 8 as the base. That is also the base of the log when we get the final answer shown above. Logs are useful to isolate the exponent.
The general rule is that [tex]\text{y} = b^{\text{x}}[/tex] is equivalent to the log form of [tex]\text{x} = \log_{b}(\text{y})[/tex] where b is the base of each.
2 = log Subscript 64 Baseline 8 is the logarithmic equation which is equivalent to 82 = 64.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
A logarithmic equation is an equation that involves the logarithm of an expression containing a variable.
The logarithmic equation equivalent to 82 = 64 is:
2 = log Subscript 64 Baseline 8
This is because the logarithmic equation relates the base (64) to the result of the exponentiation (8), which in this case is 2.
Hence, 2 = log Subscript 64 Baseline 8 is the logarithmic equation which is equivalent to 82 = 64.
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Primer numbers list maths
Answer:
1 - 100 primes numbers
Step-by-step explanation:
2,3,5,7,11,13,17,19,23,29,31,37,41,47,57,59,61,67,71,79,83,8797
The transformation from parallelogram LMNO to parallelogram L'M'N'O' was carried out as shown in the coordinate plane below.
The transformation from parallelogram LMNO to parallelogram L'M'N'O' is a reflection across the x-axis
How to determine the transformation?The complete question is added as an attachment
From the graph, we have the following highlights:
Parallelogram LMNO and parallelogram L'M'N'O' are on either sides of the x-axisThey are equidistant from the x and y axesThey have equal dimensionsThis means that parallelogram LMNO is reflected across the x-axis to get parallelogram L'M'N'O'
Hence, the transformation from parallelogram LMNO to parallelogram L'M'N'O' is a reflection across the x-axis
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Let a and b be real numbers where a b 0. Which of the following functions could represent the graph below?
A function which could represent the graph is: C. f(x) = x⁴(x - a)(x - b)².
How to interpret the graph?The nature of the intercept on a polynomial graph highlights the nature of its multiplicity. Also, the polynomial graph has no single zero (factor) but bounced off the x-axis at three (3) different locations.
This ultimately implies that the polynomial graph has an even multiplicity at these three (3) points:
x⁴
x - a
(x - b)²
In conclusion, a function which could represent the graph is f(x) = x⁴(x - a)(x - b)².
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The final result after substituting a value into an equation is the output.
A. true
B. false
I can't find the answer for 46.3 divided by 1.5 Help!
The solution to the problem when [tex]46.3[/tex] divided by [tex]1.5[/tex] the result of the division is [tex]30.8666667[/tex] in decimal form.
When a large number is distributed into small numbers of equal parts is called division.
The number [tex]1.5[/tex] is not a factor of [tex]46.3[/tex] , on dividing them the remainder is not zero.
Therefore, first convert [tex]1.5[/tex] it into a whole number by multiplying it [tex]10[/tex] so it becomes [tex]15[/tex] .
Then convert [tex]46.3[/tex] it into a whole number by multiplying it by [tex]10[/tex] so it becomes [tex]463[/tex]
Here, [tex]\dfrac{463 \times 10}{15 \times 10}[/tex] which results in [tex]\dfrac{463}{15}[/tex].
Now, divide [tex]\dfrac{463}{15}[/tex] , the result will be [tex]30.8666667[/tex].
The solution to the problem when [tex]46.3[/tex] divided by [tex]1.5[/tex] the result is [tex]30.8666667[/tex].
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first prime factor....
Answer:
The first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Which expression is equivalent to
xay
8
700
8√√√x
y
8√√y
Mark this and return
128x56
√ 2x75
? Assume x > 0 and y> 0.
Save and Exit
The equivalent expression of [tex]\sqrt{\frac{128x^5y^6}{2x^7y^5}}[/tex] is [tex]\frac{8\sqrt y}{x}[/tex]
How to determine the equivalent expression?The expression is given as:
[tex]\sqrt{\frac{128x^5y^6}{2x^7y^5}}[/tex]
Divide 128 by 2
[tex]\sqrt{\frac{64x^5y^6}{x^7y^5}}[/tex]
Apply the law of indices to the variables
[tex]\sqrt{\frac{64y^{6-5}}{x^{7-5}}}[/tex]
Evaluate the differences
[tex]\sqrt{\frac{64y}{x^2}}[/tex]
Take the square root of 64
[tex]8\sqrt{\frac{y}{x^2}}[/tex]
Take the square root of x^2
[tex]\frac{8\sqrt y}{x}[/tex]
Hence, the equivalent expression of [tex]\sqrt{\frac{128x^5y^6}{2x^7y^5}}[/tex] is [tex]\frac{8\sqrt y}{x}[/tex]
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just need b1 and b2
brainliest to whoever answers
50 points
Step-by-step explanation:
b1) Any line parallel to the x-axis is horizontal. If we look at the graph of a horizontal line, we see that for any x-value you give, the y-value will be the same. In this case, the y-value is -1. Hence, the equation for the line is [tex]y=-1[/tex], as y will be -1 no matter the x.
The gradient for a horizontal line is 0, as the "rise" of the function is 0. If we use the formula [tex]\frac{rise}{run}[/tex], we would have 0 on the top, which makes the whole fraction 0.
b2) Any line parallel to the y-axis is vertical. If we look at the graph of a vertical line, we see that for any y-value, the x-value will be the same. In this case, the x-value is -1. Hence, the equation for the line is [tex]x=-1[/tex], as x will be -1 no matter the y.
The gradient for a vertical line is undefined, as the "run" of the function is 0. If we use the formula [tex]\frac{rise}{run}[/tex], we would be dividing by 0, which is undefined.
A tape manufacturer reduces the price of its Heavy Duty tape from $40 to $38 a reel and the price of Regular tape from $32 to $31 a reel. A computing center normally spends $1600 a month for tapes and 3/5 of this is for Heavy Duty tapes. How much will they save a month under the new prices?
Answer:
68
Step-by-step explanation:
Lets first find the quantity of tape the computing center is purchasing/month.
3/5 Of their 1600 dollar spending is on the heavy duty tapes, meaning they are spending 960 dollars/month on heavy duty tape, if we divide that by the original price, 40, it will give us the original quantity of heavy tape they are purchasing, 24/month.
The other 2/5 of their money, 640 dollars, are spent on regular tape, meaning they purchase 20 rolls of regular tape/month.
Given they are purchasing 24 rolls of heavy tape and 20 rolls of regular tape every month, we can simply multiple those by the new prices and sum the results.
24*38=912
20*31=620
912+620=1532
Then to find the money they are saving we subtract the new price from the original price.
1600-1532=68
Sometimes, terminology such as "the odds are 1 in 10" of an event is used. This is interpreted to mean "the probability of the event is 1/10" or "the odds in favor of the event are 1 to 9"
The odds that a member of a certain country's Senate (selected randomly) has a law degree is 1 in 1.4. What is the probability that a member of the Senate (selected randomly) has a law degree?
The probability is
Answer:
71.4% ( i think)
Step-by-step explanation:
If we translate 1 in 1.4 using what is described, it turns into (1/1.4)
Putting this into a calculator, we get 0.714286. Translated into a percent, that becomes 71.4286%
The probability that a member of the Senate (selected randomly) has a law degree is approximately 0.4167 or 41.67%.
When the odds are given as "1 in 1.4," it means that there is 1 chance of success (having a law degree) for every 1.4 chances of failure (not having a law degree).
To find the probability that a member of the Senate (selected randomly) has a law degree, you can use the formula:
Probability = 1 / (1 + Odds)
In this case, the odds are 1 in 1.4, so the probability is:
Probability = 1 / (1 + 1.4) ≈ 1 / 2.4 ≈ 0.4167
So, the probability that a member of the Senate (selected randomly) has a law degree is approximately 0.4167 or 41.67%.
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Express v=
(2,1,5) as a linear combination of (1,2,1), (1,0,2) and (1,1,0)
The expression of v = (2,1,5) as a linear combination of (1,2,1), (1,0,2) and (1,1,0) is [tex](2,1,5)=1(1,2,1)+2(1,0,2)+-1(1,1,0)[/tex]
Linear CombinationsTo express v= (2,1,5) as a linear combination of (1,2,1), (1,0,2) and (1,1,0), it will be written as:
[tex](2,1,5)=x(1,2,1)+y(1,0,2)+z(1,1,0)[/tex]
This will yield the following equations:
x + y + z = 2........(1)
2x + z = 1..........(2)
x + 2y = 5........(3)
The solution to the system of equations above is:
x = 1, y = 2, z = -1
Therefore, the linear combination is:
[tex](2,1,5)=1(1,2,1)+2(1,0,2)+-1(1,1,0)[/tex]
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Which is true regarding the system of equations?
6 x + 2 y = 46. 3 x + y = 23.
The system results in a false statement.
The system results in an intersection at one point.
The system results in parallel lines.
The system results in a true statement because they are the same line.
Answer:
they are the same line
Step-by-step explanation:
if you divide the first line by 2 you'll get:
2(3x)+2y=2(23)===> 3x+y=23
please help I've been trying to figure it out but I can't
answer : 2 1/2
divide y by x
divide 2 1/2 by 1
its 2 1/2
10 divided by 4 is also
2 1/2
17 1/2 divided by 7 is also
2 1/2
Identify the vertices of the feasible region and use them to find the maximum and/or minimum value for the given linear programming constraints.
System of Linear Programming:
z=−3x+5y
x+y≥−2
3x−y≤2
x−y≥−4
Maximum value of z:
Minimum value of z
The maximum value of the objective function is 26 and the minimum is -10
How to determine the maximum and the minimum values?The objective function is given as:
z=−3x+5y
The constraints are
x+y≥−2
3x−y≤2
x−y≥−4
Start by plotting the constraints on a graph (see attachment)
From the attached graph, the vertices of the feasible region are
(3, 7), (0, -2), (-3, 1)
Substitute these values in the objective function
So, we have
z= −3 * 3 + 5 * 7 = 26
z= −3 * 0 + 5 * -2 = -10
z= −3 * -3 + 5 * 1 =14
Using the above values, we have:
The maximum value of the objective function is 26 and the minimum is -10
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Please help ASAP
What of the secants below is a secant?
The secant in the diagram is [tex]\overline{AD}[/tex]
Segment AD intersect the circle at two points. So it can be a secant.
We have been given that a circle of center O. We have to find the segment that is secant.
What is Secant?Secant : If a line which intersect the circle at two points.Then the line segment is called secant.
If the ray lies onto the considered line segment(overlapping), then it can be said belonging to the considered line segment.
(line segment has 2 fixed ends, whereas a ray has one end only, other end is not fixed)
Segment OB intersect the circle at one point B only. so OB cannot be a Secant.
Segment AD intersect the circle at two points. So it can be a secant.
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do you have a picture of what the graph would look like?
The attached graph represents the piecewise function.
How to determine the graph of the equation?The piecewise function is given as:
f(x) = 3x - 5, x ≤ -1
= -2x + 3, -1 < x < 4
= 2, x ≥ 4
To plot the graph, we simply plot the equations together with their domains
i.e.
3x - 5 with the domain of x ≤ -1, -2x + 3 with the domain of -1 < x < 4 and 2 with the domain of x ≥ 4
See attachment for the graph of the function
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3. A rare species of aquatic insect was discovered in the Amazon rainforest. To protect the species, environmentalists declared the insect endangered and transplanted the insect to protected area. The population P(t) (in thousands) of insects in t months after being transplanted is
a. [3 pts] Determine the number of months until the insect population reaches 40 thousand (round to 2 decimal places).
b. [3 pts] What is the limiting factor on the insect population as time progresses? Explain your answer.
c. [3 pts] Sketch a graph of the function using the window and. Be sure to indicated the scale on the graph, label the axes, at least 2 points on the graph, and any asymptotes.
The number of months until the insect population reaches 40 thousand is 14.29 months and the limiting factor on the insect population as time progresses is 250 thousands.
Given that population P(t) (in thousands) of insects in t months after being transplanted is P(t)=(50(1+0.05t))/(2+0.01t).
(a) Firstly, we will find the number of months until the insect population reaches 40 thousand by equating the given population expression with 40, we get
P(t)=40
(50(1+0.05t))/(2+0.01t)=40
Cross multiply both sides, we get
50(1+0.05t)=40(2+0.01t)
Apply the distributive property a(b+c)=ab+ac, we get
50+2.5t=80+0.4t
Subtract 0.4t and 50 from both sides, we get
50+2.5t-0.4t-50=80+0.4t-0.4t-50
2.1t=30
Divide both sides with 2.1, we get
t=14.29 months
(b) Now, we will find the limiting factor on the insect population as time progresses by taking limit on both sides with t→∞, we get
[tex]\begin{aligned}\lim_{t \rightarrow \infty}P(t)&=\lim_{t \rightarrow \infty}\frac{50(1+0.05t)}{2+0.01t}\\ &=\lim_{t \rightarrow \infty}\frac{50(\frac{1}{t}+0.05)}{\frac{2}{t}+0.01}\\ &=50\times \frac{0.05}{0.01}\\ &=250\end[/tex]
(c) Further, we will sketch the graph of the function using the window 0≤t≤700 and 0≤p(t)≤700 as shown in the figure.
Hence, when the population P(t) (in thousands) of insects in t months after being transplanted by P(t)=(50(1+0.05t))/(2+0.01t) then the number of months until the insect population reaches 40 thousand 14.29 months and the limiting factor on the insect population is 250 thousand and the graph is shown in the figure.
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