The object will have a height of 100 feet after 100 seconds.
The equation for the height of the object in feet after T seconds is H(T) = A(T - h)2 + k, where A and h are constants. Given that the object has a height of 0 feet after 10 seconds, and a maximum height of 200 feet at 0 seconds, we can solve for A and h.
We know that H(0) = 200, so k = 200.
We also know that H(10) = 0, so A(10 - h)2 + 200 = 0. Solving for A, we get A = -(h2 + 200) / 102.
Now we have an equation with two unknowns, A and h, so we can use substitution to solve for h. We can substitute -(h2 + 200) / 102 for A in our original equation, and solve for h. This gives us h = 10.
We now know A = -(102 + 200) / 102 = -20.
So, our equation for the height of the object in feet after T seconds is H(T) = -20(T - 10)2 + 200.
To find the height of the object after 100 seconds, substitute 100 for T in the equation. This gives us H(100) = -20(100 - 10)2 + 200 = 100.
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unit linear relationships student handout 1 answer key
In statistics, a linear relationship is a connection that runs in a straight line between two variables (or linear association).
What do you mean by linear relationships?Both a mathematical representation of a linear relationship, in which the dependent variable is produced by multiplying the independent variable by the slope coefficient and a constant, and a graphic representation of a linear relationship, in which the variable and constant are connected by a straight line.
A polynomial or non-linear (curved) relationship can be contrasted with a linear relationship.
A linear relationship is one that mathematically answers the question:
y = mx + b
Where,
m = slope
b = y-intercept
"X" and "Y" are two variables in this equation.
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Find the measurements of X
pt
Answer:
x = 15°
Step-by-step explanation:
the inscribed angle APB is half the measure of the central angle AOB , so
x = [tex]\frac{1}{2}[/tex] × 30° = 15°
Maricopa's Success scholarship fund receives a gift of $ 85000. The money is invested in stocks, bonds, and CDs. CDs pay 5.5 % interest, bonds pay 5.2 % interest, and stocks pay 6.4 % interest. Maricopa Success invests $ 20000 more in bonds than in CDs. If the annual income from the investments is $ 4780 , how much was invested in each account?
Maricopa Success invested $ in stocks.
Maricopa Success invested $ in bonds.
Maricopa Success invested $ in CDs.
To solve this problem, we can use a system of equations. Let's represent the amount invested in stocks as x, the amount invested in bonds as y, and the amount invested in CDs as z.
The first equation we can create is the total amount invested:
x + y + z = 85000
The second equation is the difference between the amount invested in bonds and CDs:
y - z = 20000
The third equation is the total interest earned:
0.064x + 0.052y + 0.055z = 4780
We can use substitution to solve this system of equations. Let's rearrange the second equation to solve for y:
y = z + 20000
We can then substitute this value of y into the first equation:
x + (z + 20000) + z = 85000
Simplifying gives us:
x + 2z = 65000
Now, let's substitute the value of y into the third equation:
0.064x + 0.052(z + 20000) + 0.055z = 4780
Simplifying gives us:
0.064x + 0.107z = 3780
We can now use the first and third equations to solve for x and z. Let's rearrange the first equation to solve for x:
x = 65000 - 2z
We can then substitute this value of x into the third equation:
0.064(65000 - 2z) + 0.107z = 3780
Simplifying gives us:
4160 - 0.128z + 0.107z = 3780
Combining like terms gives us:
-0.021z = -380
Solving for z gives us:
z = 18095.24
We can now use this value of z to solve for y:
y = z + 20000 = 18095.24 + 20000 = 38095.24
And finally, we can use the value of z to solve for x:
x = 65000 - 2z = 65000 - 2(18095.24) = 28809.52
So, Maricopa Success invested $28809.52 in stocks, $38095.24 in bonds, and $18095.24 in CDs.
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600% of what number is 2,280
Answer:
380
Step-by-step explanation:
To find out what number is 600% of another number, we need to divide that number by 600% (or 6).
So:
x = 2,280 ÷ 6
x = 380
Therefore, 600% of 380 is equal to 2,280.
Hope this helped !
Dividing a polynomial by a monomial: (-15xz+18x^(6)z^(7)+25x^(5)z^(2))-:(-3x^(5)z^(2)) mplify your answer as much as possible.
The simplest form obtained by the given division is (-25 - 18x^(5)z^(6) + 15x^(4)z)/(3x^(5)z^(2)) .
To divide a polynomial by a monomial, we simply divide each term in the polynomial by the monomial. So, we will divide each term in (-15xz+18x^(6)z^(7)+25x^(5)z^(2)) by (-3x^(5)z^(2)).
First, we will divide -15xz by -3x^(5)z^(2):
-15xz/(-3x^(5)z^(2)) = 5x^(-4)z^(-1)
Next, we will divide 18x^(6)z^(7) by -3x^(5)z^(2):
18x^(6)z^(7)/(-3x^(5)z^(2)) = -6x^(1)z^(5)
Finally, we will divide 25x^(5)z^(2) by -3x^(5)z^(2):
25x^(5)z^(2)/(-3x^(5)z^(2)) = -25/3
So, our final answer is:
5x^(-4)z^(-1) - 6x^(1)z^(5) - 25/3
In simplified form, this is:
(-25 - 18x^(5)z^(6) + 15x^(4)z)/(3x^(5)z^(2))
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Use the table to write a linear function that relates to y to x. PLEASEE HELP
to get the equation of any straight line, we simply need two points off of it, let's use those two points in the picture below.
[tex](\stackrel{x_1}{3}~,~\stackrel{y_1}{7})\qquad (\stackrel{x_2}{6}~,~\stackrel{y_2}{9}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{9}-\stackrel{y1}{7}}}{\underset{\textit{\large run}} {\underset{x_2}{6}-\underset{x_1}{3}}} \implies \cfrac{ 2 }{ 3 }[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{ \cfrac{ 2 }{ 3 }}(x-\stackrel{x_1}{3}) \\\\\\ y-7=\cfrac{ 2 }{ 3 }x-2\implies {\Large \begin{array}{llll} y=\cfrac{ 2 }{ 3 }x+5 \end{array}}[/tex]
A caterer made 24 sandwiches for a company lunch. When they arrived at the location. they discovered the guest list was more than they had anticipated. They decided to cut the sandwiches into ( l)/(3) size pleces. How many ( 1)/(3) size pieces were they able to make?
The caterer was able to make 72 (1)/(3) size pieces from the 24 sandwiches.
To find this answer, we can use the following equation:
24 sandwiches × 3 (1)/(3) size pieces per sandwich = 72 (1)/(3) size pieces
This equation tells us that for every sandwich, the caterer was able to make 3 (1)/(3) size pieces. By multiplying the number of sandwiches by the number of (1)/(3) size pieces per sandwich, we can find the total number of (1)/(3) size pieces the caterer was able to make.
Therefore, the caterer was able to make 72 (1)/(3) size pieces from the 24 sandwiches.
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Stella's family traveled
3
10
of the distance to her aunt’s house on Saturday. They traveled
4
7
of the remaining distance on Sunday. What fraction of the total distance to her aunt’s house was traveled on Sunday?
Stella's family traveled 2/5 of the total distance to her aunt's house on Sunday.
What is distance ?
Distance is a physical quantity that refers to the length between two points or the amount of space between them. It is typically measured in units such as meters, kilometers, miles, or feet.
Stella's family traveled 3/10 of the distance to her aunt's house on Saturday. Therefore, the remaining distance they had to travel was:
1 - 3/10 = 7/10
On Sunday, they traveled 4/7 of the remaining distance. Therefore, the total distance traveled on Sunday is:
(4/7) x (7/10) = 4/10
Simplifying the fraction 4/10 gives 2/5.
Therefore, Stella's family traveled 2/5 of the total distance to her aunt's house on Sunday.
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(−6, 3), (−6, 5), (2, 3)
What are the coordinates of the fourth vertex of the rectangle
The coordinates of the fourth vertex of the rectangle are (2, 5).
To find the fourth vertex of the rectangle, we need to first determine which sides of the triangle are opposite and parallel to each other. We can see that the two points (-6,3) and (-6,5) lie on a vertical line, which means they are opposite and parallel to each other.
Similarly, the two points (-6,3) and (2,3) lie on a horizontal line, which means they are also opposite and parallel to each other. The fourth vertex of the rectangle must lie at the intersection of these two lines.
The point (-6,3) is the lower left corner of the rectangle, and the point (2,3) is the lower right corner. The distance between these two points is 2 - (-6) = 8. Since the opposite sides of a rectangle are congruent, the distance between the other two corners (-6,5) and the fourth vertex must also be 8.
Since the two points (-6,3) and (-6,5) are on a vertical line, the y-coordinate of the fourth vertex must also be 5. We can find the x-coordinate by moving 8 units to the right of (-6,3), since the rectangle is parallel to the x-axis. Thus, the fourth vertex must be located at the point:
(x,y) = (-6+8, 5)
= (2, 5)
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AB-> (2;3) AB = ?
les coordonées de mon vecteur AB-> sont ( 2;3 ) est ce que AB= racine de( 2 au carre + 3 au carée)
Answer:
Réponse :Explications étape par étapea) Le vecteur AC(xc-xa;yc-ya)vecteur AC(-6;-3
Step-by-step explanation:
Les coordonnées d'un vecteur dans un r.o.n. décrivent le déplacement qu'il représente. Ainsi, un déplacement de « 3 ...
The graph shows the number of prints Tara needs to sell to make a profit. What can you learn by looking at the graph? IMAGINE MATH
This graph shows that Tara’s prints have the potential to be profitable, but will require a significant amount of sales to make a profit.
What is profit?Profit is the difference between a firm's total revenue and total expenses. It is the amount of money a business has earned after subtracting the cost of goods sold, operating expenses and taxes from its total revenue.
The graph reveals a few key pieces of information that can be used to assess the potential profitability of Tara’s prints. Firstly, the graph shows that there is a fixed cost associated with each print, regardless of how many prints are sold. This cost is represented by the straight line at the bottom of the graph. Additionally, the graph reveals that the cost of the prints increases exponentially as more prints are sold. This means that the more prints Tara sells, the more money she makes. Lastly, the graph shows that the revenue from selling prints is significantly higher than the cost of producing them, indicating that Tara can make a profit if she is able to sell enough prints. Overall, this graph shows that Tara’s prints have the potential to be profitable, but will require a significant amount of sales to make a profit.
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HELP NOW PLSSS
Solve. -7 2/3 + (-5 1/2) + 8 3/4 =
Answer:
-4.4166
Step-by-step explanation:
-23/3+(-11/2)+35/4
-24/3-11/2+35/4
use bodmas
addition first then subtract and u find your LCM.
11/2+35/4=57/4 or 14 1/4
LCM=4
-23/3-57/4=-263/12 or -21 11/12.
LCM=12
final answer= -21 11/12.
Initially, there were only 3 weeds at a park. The weeds grew at a rate of 5% each week. The following function represents the weekly weed growth: f(x) = 3(1.05)x. Rewrite the function to show how quickly the weeds grow each day and calculate this rate as a percentage.
Answer: To rewrite the function to show how quickly the weeds grow each day, we need to use the fact that there are 7 days in a week. Let's divide both sides of the original function by 7 to get:
f(x/7) = 3(1.05)^(x/7)
This function gives the amount of weed growth after x/7 weeks, which is equivalent to x days. To find the daily weed growth rate, we need to take the derivative of this function with respect to x and evaluate it at x=0:
f'(0) = (d/dx) 3(1.05)^(x/7)
Using the chain rule, we get:
f'(0) = 3ln(1.05)/7 ≈ 0.00797
This means that the weeds grow at a daily rate of approximately 0.797%, or 0.00797 as a decimal.
Step-by-step explanation:
Which algebraic expression represents this word description the product of seven and the diffrence between a number and ten
The algebraic expression represents this word description is 7(x - 10) (option D).
One way to approach this problem is to break down the phrase into its components and then use mathematical symbols to represent them.
The phrase contains the word "product," which tells us we need to multiply two quantities.
The first quantity is the number seven, and the second quantity is the difference between a number and ten.
To represent the difference between a number and ten, we can use the variable x to stand for the unknown number.
Then, the difference between x and ten can be expressed as (x - 10).
Putting it all together, the expression that represents the product of seven and the difference between a number and ten is 7(x - 10).
Hence the correct option is (D).
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Complete Question:
Which algebraic expression represents this word description?
The product of seven and the difference between a number and ten
A. 10-7x
B. 7(10 - x)
C. 7x-10
D. 7(x - 10)
Combine like terms and write the resulting polynomial in descending 2x^(7)+5x^(6)+9x^(7) Select one:
The resulting polynomial in descending order is 11x7 + 5x6
To combine like terms and write the resulting polynomial in descending order, we need to first identify the like terms and then add or subtract their coefficients. Like terms are terms that have the same variable and the same exponent.
In this case, the like terms are 2x^(7) and 9x^(7).
We can combine these like terms by adding their coefficients:
2x^(7) + 9x^(7) = 11x^(7)
Now we have:
11x^(7) + 5x^(6)
Since the exponents are different, we cannot combine these terms.
Therefore, the resulting polynomial in descending order is 11x7 + 5x6
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What is the area of the parallelogram?
37 in.
31 in.
425.5 in.²
713 in. 2
851 in.2
1147 in.²
23 in.
Area of the parallelogram, using the area formula = 851 in².
What is the area of a parallelogram?The area of a parallelogram, which is expressed in square units, is the total number of unit squares that may fit inside of it (like cm², m², in² etc). It is the region that a two-dimensional parallelogram encloses or encompasses.
Hence, the following formula can be used to determine a parallelogram's area:
Area of a parallelogram is equal to b × h square units, where b is the base's length and h is its height.
Here in the question,
b = 37in.
h = 23in.
Area = b × h
= 37 × 23
= 851 in²
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Simplify.
4x^3 - 12x^2
__________
4x^2 + 7x - 2
__________
2x^2 - 6x
__________
5x^2 + 11x + 2
Answers:
A. 2x(5x-1)
_____
4x-1
B. 2x(5x+1)
_____
4x-1
C. x(5x+1)
_____
4x-1
D. x(5x+1)
_____
2(4x-1)
Answer: 7 .
Step-by-step explanation:
The middle term is, +7x its coefficient is 7 . Step-2 : Find two factors of -8 whose sum equals the coefficient of the middle term, which is
If no one helps me on this, I will get a zero :(
Answer:
Step-by-step explanation:
What is the multiplicity of the zero x=1 for the function f(x)=x^(2)(x-1)^(4)(x+5)
The multiplicity of the zero [tex]x=1[/tex] for the function [tex]f(x)=x^2(x-1)^(4)(x+5)[/tex] is 4.
In a polynomial function, the multiplicity of a zero is the number of times that zero appears as a factor in the function. In other words, it is the exponent of the factor corresponding to that zero.
In the given function [tex]f(x)=x^2(x-1)^4(x+5)[/tex], we can see that the zero x=1 corresponds to the factor (x-1)^(4). This means that the multiplicity of the zero x=1 is 4, as it appears as a factor 4 times in the function.
Therefore, the multiplicity of the zero [tex]x=1[/tex] for the function [tex]f(x)=x^2(x-1)^(4)(x+5)[/tex] is 4.
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The points P(7, -3), Q(0,3), R(-9,5), and S(-2,-1) form a quadrilateral. Find the desired slopes and lengths, then fill in the words that BEST identifies the type of quadrilateral.
The quadrilateral formed from the points P(7, -3), Q(0,3), R(-9,5), and S(-2,-1) is a rhombus.
To find the desired slopes and lengths, we can use the slope formula and the distance formula. The slope formula is (y₂ - y₁)/(x₂ - x₁) and the distance formula is √((x₂ - x₁)² + (y₂ - y₁)²).
Slope of PQ = (3 - (-3))/(0 - 7) = 6/(-7) = -6/7
Slope of QR = (5 - 3)/(-9 - 0) = 2/(-9) = -2/9
Slope of RS = (-1 - 5)/(-2 - (-9)) = (-6)/(7) = -6/7
Slope of SP = ((-3) - (-1))/(7 - (-2)) = (-2)/(9) = -2/9
Length of PQ = √((0 - 7)² + (3 - (-3))²) = √(49 + 36) = √85
Length of QR = √((-9 - 0)² + (5 - 3)²) = √(81 + 4) = √85
Length of RS = √((-2 - (-9))² + ((-1) - 5)²) = √(49 + 36) = √85
Length of SP = √((7 - (-2))² + ((-3) - (-1))²) = √(81 + 4) = √85
Since the opposite slopes are equal and all of the lengths are equal, the quadrilateral is a rhombus.
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HELP 75PTS!!!!!! Explain how to evaluate 34.
(Write 3 or 4 sentences)
Answer:
Now, This number is neither a perfect square nor a perfect cube
So, 34 can be evaluated as, 34 = (30 + 4) (15 + 15 + 4)
Here is the evaluated form of 34.
Step-by-step explanation:
is of goggle btw so just change some of the words
1. Find all real solution(s) to each equation algebraically: (a) - n + √(6n + 19) = 2 (Remember to check your solutions.) (b) x^4 - 20x^2 + 64 = 0
the real solutions to this equation are x = 2√2 and x = -2√2.
Solution:
(a) -n + √(6n + 19) = 2
To find the solution algebraically, we need to isolate the radical on one side of the equation and then square both sides to get rid of the radical.
-n = 2 - √(6n + 19)
(-n - 2)^2 = (2 - √(6n + 19))^2
n^2 + 4n + 4 = 4 - 4√(6n + 19) + 6n + 19
n^2 - 2n - 19 = -4√(6n + 19)
(n^2 - 2n - 19)^2 = (-4√(6n + 19))^2
n^4 - 4n^3 - 18n^2 + 76n + 361 = 96n + 304
n^4 - 4n^3 - 114n^2 + 76n + 57 = 0
This is a quartic equation that can be solved using the Rational Root Theorem or by factoring. However, this equation does not have any real solutions. Therefore, there are no real solutions to this equation.
(b) x^4 - 20x^2 + 64 = 0
This equation can be factored as:
(x^2 - 8)^2 = 0
x^2 - 8 = 0
x^2 = 8
x = ±√8
x = ±2√2
Therefore, the real solutions to this equation are x = 2√2 and x = -2√2.
Remember to check your solutions by plugging them back into the original equation and seeing if they make the equation true.
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In a garment factory, a women work b hours a day and produce c articles each day. If
d women are released, how many hours a day will the remaining women have to work
to produce c articles each day? (There’s an attached photo to this !)
The equation (bd)/(b-d) = c can be used to determine how many hours a day the remaining women must work in order to produce the same number of articles each day, when some of the women have been released.
What is an equation?An equation is an expression used to relate two or more quantities, usually by using mathematical operators such as addition, subtraction, multiplication and division. An equation typically has an equal sign (=) as its central element, with the left and right sides of the equation representing the two sides of the equation.
The answer to this question can be determined by using the following equation: (bd)/(b-d) = c. This equation is a simple ratio that takes into account the number of hours the women work per day (b), the number of women released (d), and the number of articles produced each day (c).
For example, if a factory had 10 women working 8 hours a day and produced 100 articles each day, and 5 of those women were released, then the equation (8 x 5) / (8-5) = 100 would determine that the remaining women must work 12.5 hours per day in order to produce the same number of articles.
In conclusion, the equation (bd)/(b-d) = c can be used to determine how many hours a day the remaining women must work in order to produce the same number of articles each day, when some of the women have been released.
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The scale factor of two similar hexagons is 3:7.
The area of the smaller hexagon is 18 m².
What is the area of the larger hexagon?
36 m²
42 m²
98 m²
5832 m²
324 m²
Answer:
= 42m²
Step-by-step explanation:
The ratio of two similar hexagons is 3:7
where the smaller correspond to 3
And
the bigger correspond to 7
where the area of the smaller haxagon is 18
so if 3 = 18m²
then 7 = ?
therefore
[tex] \frac{7}{3} \times 18 \\ = 42 {m}^{2} [/tex]
therefore the area of the bigger hexagon is 42m²
Find all the missing angels
Answer:
15, 15
Step-by-step explanation:
Since this triangle is a right triangle has it's other two angles at 45 degrees, (180-90-45=45 for the last angle) we know that it's a 45-45-90 triangle and can use the rules in the image below to solve the other two sides:
Because the hypotenuse of this triangle is 15[tex]\sqrt{2}[/tex], then we know that the other two side lengths are 15[tex]\sqrt{2}[/tex]/[tex]\sqrt{2}[/tex], or in this case, just 15.
Answer: 15, 15
Step-by-step explanation:
It is a 45, 45, 90 triangle. This means that both of the legs have the same lengths and the hypotenuse is sqrt2 times the length of one of the legs.
Since sqrt2 is irrational, it is easy to look at the coefficient of it. Since the coefficient is 15, the lengths of the sides are also 15.
I am a function. My parent function is y = x. My parent function is
mapped onto me by a reflection over the line y = 0, then a horizontal shift
4 units to the right, a vertical shift 3 units up, and finally a horizontal
stretch with a factor of 3. Who am I?
Answer:
Step-by-step explanation:
If the graph of the parent function is shifted 7 units up and is steeper than the parent function, which of these could represent the function? answer choices.
NEED HELP (IMAGE BELOW)
Lina hops linearly and Kavier hops exponentially
The equations of the functions are Lina: y = 2x + 3 and Kavier: y = 3(2)^x
The pattern of loops of the KangaroosLinear pattern
The table of values represent the given parameter
For Lina's distance from the tree, we can see that
As the hop increases by 1, the distance covered increases constantly by 2 feet
This represents a linear function
Exponential pattern
For Kavier's distance from the tree, we can see that
As the hop increases by 1, the distance covered doubles
This represents an exponential function
How to determine the equations
Lina Linear Function
From the question, we have the following parameters that can be used in our computation:
(0, 3) and (1, 5)
A linear function is represented as
y = mx + c
Where
c = y when x = 0
So, we have
y = mx + 3
Using the points, we have
m + 3 = 5
m = 2
So, we have
y = 2x + 3
Kavier Exponential Function
Here, we have
(0, 3) and (1, 6)
An exponential function is represented as
y = ab^x
Where
a = y when x = 0
So, we have
y = 3b^x
Using the points, we have
3b = 6
b = 2
So, we have
y = 3(2)^x
Hence, Kavier's equation is y = 3(2)^x
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Pls give simple working
Answer:
132, 115
Step-by-step explanation:
For the first one:
Bsc its parallelogram angle x is 132
Second one:
x + 108 + 68 + 59 = 360
x + 235= 360
x = 360 - 235
x = 115
find the value of largest rectangular parallelopiped that can be incribed in the ellipsoid
The value of the largest rectangular parallelopiped that can be inscribed in the ellipsoid is (2a)*(2b)*(2c).
To find the value of the largest rectangular parallelopiped that can be inscribed in the ellipsoid, we need to use the formula for the volume of a parallelopiped. The formula is:
V = l*w*h
Where V is the volume, l is the length, w is the width, and h is the height.
Since we are looking for the largest rectangular parallelopiped, we need to maximize the volume. This can be done by finding the maximum values of the length, width, and height.
The maximum values of the length, width, and height can be found by using the semi-axes of the ellipsoid. The semi-axes of the ellipsoid are given by:
a = √(x^2/a^2 + y^2/b^2 + z^2/c^2)
Where a, b, and c are the semi-axes of the ellipsoid, and x, y, and z are the coordinates of the point on the surface of the ellipsoid.
The maximum values of the length, width, and height can be found by setting the derivatives of the volume with respect to x, y, and z to zero:
∂V/∂x = 0
∂V/∂y = 0
∂V/∂z = 0
Solving these equations will give us the maximum values of the length, width, and height. Once we have these values, we can plug them into the formula for the volume of a parallelopiped to find the largest volume.
The largest volume of the rectangular parallelopiped that can be inscribed in the ellipsoid is given by:
V = (2a)*(2b)*(2c)
Where a, b, and c are the semi-axes of the ellipsoid.
Therefore, the value of the largest rectangular parallelopiped that can be inscribed in the ellipsoid is (2a)*(2b)*(2c).
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Consider the decision sapling in which you can choose either a sure thing worth $5M or a gamble that has probability of 0.4 of giving a return of $2M; otherwise the return on the gamble will be $7M. Suppose we can purchase a forecast for the gamble, where the forecast has probability p of being accurate. (So the false positive and false negative rates are both 1-p). a. What is the EMV of the gamble given the forecast is that the return will be $7M? If it is $2M? b. If the forecast is $7M, for what range of p will it be optimal to choose the gamble? c. Give an expression for the EVII as a function of p, and graph it.
The EMV (Expected Monetary Value) of the gamble is calculated as follows:
EMV = (probability of return) x (value of return)
a. If the forecast is that the return will be $7M, then the EMV of the gamble is:
EMV = (0.4) x ($7M) = $2.8M
If the forecast is that the return will be $2M, then the EMV of the gamble is:
EMV = (0.6) x ($2M) = $1.2M
b. If the forecast is $7M, the range of p for which it will be optimal to choose the gamble is when the EMV of the gamble is greater than the sure thing of $5M. This can be calculated by setting the EMV of the gamble equal to $5M and solving for p:
$5M = (0.4) x ($7M) + (1-p) x ($2M)
$5M = $2.8M + $2M - $2Mp
$2.2M = $2Mp
p = 1.1
Therefore, the range of p for which it will be optimal to choose the gamble is when p > 1.1.
c. The EVII (Expected Value of Imperfect Information) is the difference between the EMV of the gamble with the forecast and the EMV of the gamble without the forecast. This can be expressed as a function of p as follows:
EVII = (p x EMV with forecast) - (1-p) x EMV without forecast
EVII = (p x ($2.8M)) - (1-p) x ($1.2M)
EVII = $2.8Mp - $1.2M + $1.2Mp
EVII = $4Mp - $1.2M
To graph this function, plot p on the x-axis and EVII on the y-axis. The slope of the line will be $4M and the y-intercept will be -$1.2M.
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