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If Utility Function is Expressed as U(x,y) = x^0.5 y^0.5 What is the Marginal Utility at Point (64,25) and (49,36)? Treat Y as a Constant. Would the Answer be MUx(x,y) = y. MUx(64,25) = 5 ? and MUx(49,36) = 6?

Solution:

MUx(x,y) = Derivative of U(x,y), treating y constant.

MUx(64,25) =

MUx(49,36) =

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Is 288/1155 in Lowest Terms

Solution:

First, factorize the numerator using prime numbers: 288 = 2*12*12 = 2*3*4*3*4=2*2*2*2*2*3*
Second, factorize the denominator: 1155 = 5*231=5*7*33 = 5*7*3*11
Notice that the numerator and denominator have only one factor in common (3).
Restate the ratio after removing the common factor
1. New numerator = 2*2*2*2*2*3 = 96
2. New denominator = 5*7*11 = 385
3. Same ratio in lowest terms:

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Factor These: 2w^2-5w-10

Solution:

We can factor the expression as follows:

To figure this out, we used the quadratic formula (see http://en.wikipedia.org/wiki/Quadratic_equation):

w =

We recognized that y= is a quadratic equation in the form of y= where a=2, b=-5 and c=-10.

To confirm that our calculation are right, we decided to plot the graph of y = by plugging in various values of w.

As you can see, there is a root at approximately -1.3 and another root at approximately +3.8. So we know approximately what the values of a and b are. We could use a calculator to get closer and closer to the exact values of a and b. For example, if we try w=3.8, we get a value of y = -0.12. If we try a slightly different value, we can see if we’re getting closer to zero or farther away: w=3.81 produces a value of y=-0.018, so we know that 3.81 is a better approximation than 3.8. If we keep calculating closer … read more

A Jeweler Needs to mix an Alloy with 16% Gold Content & an Alloy with a 28% Gold Content to Obtain 32oz. of a new Alloy with a 25% Gold Content. How Many oz. of Each of the Original Alloys Must be Used?

Solution:

A = number of ounces of the first alloy
B = number of ounces of the second all
A+B=32
0.16 A +0.28 B = 32 x 0.25
Since A + B = 32, we know that A = 32 – B. You can then re-write the equation above as 0.16 (32-B) + 0.28 B = 32 x 0.25
This can be simplified to 5.12- 0.16 B + 0.28 B =8
Or, 0.12 B = 2.88
Or B = 24
So A = 8

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Solve the Triangle. Round Angle Measures to the Nearest Minute & Side Measures to the Nearest Tenth. a=11.4, b=13.7, c=12.2

Solution:

These are the formulas used to solve triangles:

1. The sum of the internal angles equals 180^o …A + B + C = 180^o

2. The ‘sine rule’ …

3. The ‘cosine rule’ …

a² = b² + c² – 2bc cosA

b² = a² + c² – 2ac cosB

c² = b² + a² – 2ba cosC

In the problem that you gave us, we know the length of the three sides of the triangle:

a = 11.4

b = 13.7

c = 12.2

When no angles are known, the cosine rule is the only option, so 11.4² =13.7² +12.2² – 2*13.7*12.2*cosA….Therefore cosA = 0.617955.

You can now use a calculator or a table to find the value of A. You should get A = 0.9047 radians = 51.833 degrees = 51 degrees and 50 minutes.

Use the sine rule to find one of the remaining angles.

So sin(B) = = 0.944835

You can now … read more

What is the Largest Amount of Postage That Cannot be Made Using These Stamp Denominations: 17 Cents, 9 Cents & 5 Cents?

Solution:

We’re not sure we understand the question, because if you have enough stamps (of any denomination) you can cross any amount. However, you said that the total value (let’s call is “q”) has to be less than a million dollars (e.g. $999,999.99), so let’s work backwards. $999,999.99 can be obtained as follows:

5,882,350 stamps of 17 cent denomination
1 stamp of 9 cent denomination
8 stamps of 5 cent denomination

So we have shown the largest amount of postage that can be paid (below the million dollar limit) is $999,999.99.

Now let’s assume that the amount paid has to be exact, i.e. we cannot overpay. This is a lot more complicated, as to do this we have to consider all values of q, where q = 17x + 9y + 5z, and where x, y and z are positive integers. We could write a computer program to check the value of q for all values of x, y and z, and then see which … read more

-2x < 3/7

Solution:

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Cos 2x=0.32

Solution:

First let’s apply the reverse cosine function to both sides of the equation:

cos^-1(cos(2x)) = cos^-1 (0.32) = cos^-1 ()

Now if you take the inverse cosine the cosine of 2x, you get 2x…so:

2x = cos^-1 ()

2x 71.34°

x 35.67° (which is approximately the same as 0.622 radians)

This looks like it might be the final answer, but actually it’s only one of the many correct answers. One way to see this is by graphing y=cos(2x) and seeing where it intersects the line y=0.32. If you do this, you will see that the intersections occur at multiple points:

As you can see, the intersections occur at x 0.622 radians, 2.519 radians, 3.764 radians, 5.660 radians, etc. The general form of the solution set is as follows:

What is the Volume of a Sphere the Radius of Six?

Solution:

The formula for the volume of a sphere is
r = 6
=

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What is an Inverse Function?

If you have a function called ƒ, let’s call its inverse ƒ^–1 (i.e. ƒ^–1 is the inverse function of ƒ).

By definition, the property of ƒ^–1 is that if ƒ(a)=b, then ƒ^–1(b)=a. Wikipedia (http://en.wikipedia.org/wiki/Inverse_function) has a good example:

A function ƒ and its inverse ƒ^–1. Because ƒ maps a to 3, the inverse ƒ^–1 maps 3 back to a. In mathematics, if ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, with the property that a round trip from A to B to A (or from B to A to B) returns each element of the initial set to itself.… read more

Solve them by substitution method: 5b – 3a = 2, (a-1)^2 + (b-1)^2 = 34

Solution:

First, we can write

Then we can substitute this into the second equation you get

We can simplify this to

If multiply both sides by 25 and expand this out we get

This can be simplified to: Or

Thus the solutions for a are 6 and -4

The solutions for b can be found by the equation (b=4 when a=6, and b=-2 then a=-4)

Another way to solve this is by looking at the equations graphically. When you plot the two equations, you can see there they intersect.

As you can see, they intersect at two places:

You can plug the answers into both equations to confirm that they are both true using these two values of a and b.… read more

2x-5y=9 and 3x+4y=25. Solve by the method of elimination.

Solution:

1. 3x + 4y=25
2. 2x – 5y=9

We could rewrite these as follows by multiplying the first equation by 5 and the second by 4:

1. 15x + 20y = 125
2. 8x – 20y = 36

If we add the above two equations, we can eliminate y:

23x = 161

So x =3D 7, and plugging this into the first equation we can figure out
y: 21 + 4y =3D 25,

so y = 1… read more

Solve this problem by using elimination: 2x-3y=61 and 2x+y=-7

Solution:

2x – =y=61
2x + y= -7

If you subtract the second equation from the first, you can eliminate x and you are left with -4y = 68.

So y =3D -17, and plugging this into the first equation we can figure out

y: 2x = (-51)=61,

so x = 5… read more

12 out of 1000 women at age forty who participate in routine screening have breast cancer. 880 out of 1000 women with breast cancer will get positive mammographies. 90 out of 1000 women without breast cancer will also let positive mammographies. If 100 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?

Solution:

1.   Out of 100 women, we expect 1.2 to actually have breast cancer (12 out of 1000)
2.   However, only 88% (880 out of 1000) of these will be detected, so 1.056 (i.e. 88% of 1.2) will be detected
3.   In addition, of the 98.8 women who don’t have breast cancer 9% (90 out of 1000) will trigger a false positive” (9% of 98.8 = 8.892)
4.   So out of 100 women, 9.948 will have positive mammography readings (1.056+9.892)
5.   Of these 9.948, only 1.056 will actually have breast cancer
6.   10.6% = (1.056/1.056+9.892) This ratio is essentially an application of Bayes Theorem

Please click on http://www.thegodofreason.com/bayesintro.pdf to download a PDF that explains how Bayes Theorem works. Page 3 gives an example that is very similar to your problem:

Need to get the answer to this problem by using elimination: =x-3y=61 and 2x+y=-7

Solution:

2x – =y=61
2x + y= =7

If you subtract the second equation from the first, you can eliminate x and you are left with -4y = 68.

So y = 3D -17, and plugging this into the first equation we can figure out

y: 2x = (-51)=61,

so x = 5… read more

What is 9/10 of 80?

Solution:

The expression “9/10 of 80” can be written as

This is equal to 72.… read more

What is the name of a 6-sided polygon?

Solution:

A six-sided polygon is known as a Hexagon:

I had 48 marbles in my jar. I lost 1/8 and gave 1/6 to my best friend. How many marbles were left in my jar?

Solution:

Start with 48
Lost 6 (1/8 of 48 =s 6)
Gave 8 to a friend (1/6 of 48 is 8)
48 – 6 – 8
= 48 – 14
= 34

Solve equation. Round to nearest hundredth if neccessary. 11xsquared+3=5(4xsquared-3)

Solution:

The equation you gave us:
Can be re-written =s: or or
The Solution is:
This can also be written in an approximate decimal form: +1.41 and -1.41

I need help with solving systems by graphing. this is one of the problems y=5x-2 y=x+6

Solution:

First write down the two equations and then pick several values of x and use those values to calculate the corresponding values of y. You can use the x,y pairs to then plot each line. You will see that the two lines intersect at 2,8 =i.e. x=2 and y=8).

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Topics under "free math help/feed/rss2/get-started.html"

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