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Question 53

A water trough is 10m long and a cross-section has the shape of an isosceles triangle that is 1m across at the top and is 50cm high. The trough is being filled with water at the rate of 0.4m3/min. How fast will the water level rise when the water is 40cm deep?

Solution 53

As the trough is being filled, three values are changing with respect to time. The volume of the water in the trough, the height of the water, and the length of the base of the triangle. Let the base of the isosceles triangle be d, and height be h. Since it is isosceles, can use similar triangles to get d in terms of h:
d/h=1/0.5 so, d=2h

Volume of water in the trough,
V= area of the triangle*length of the trough
= (1/2)d*h*10
= 5*2h*h
=10h2

take derivative of V with respect to time:
dV/dt= 20h dh/dt (chain rule)
dh/dt= (1/20h) dV/dt

You know that the rate of water flow is 0.4 m3/min. This is also how fast the volume of water in the trough is changing, ie dV/dt.
You want to find out how fast the water level is rising (dh/dt) when the height is 40cm=0.4m (h)
so substitute dV/dt=0.4 and h=0.4 into dh/dt.

dh/dt= [1/(20*0.4)]*0.4= 1/20= 0.05 m/min= 5 cm/min
So, the water level is rising at 5 cm/min when the height is 40cm.


Question 54

For f(x) = 3x2 + x, g(x) = sqrt x, find the (g o f)(2)

Solution 54

(g o f)(x)= g[f(x)]
This means take f(x) and put it in g.
So, g[f(x)]= sqrt(3x2+x)
g[f(2)]= sqrt[3(22)+2]
= sqrt(14)


Question 55

2 log (In 6) or (36sqrt6).

Solution 55

2 log (36sqrt6)
= 2[log (36) + log (sqrt 6)]
(log of a pdt can be written as the sum of logs.)
= 2 log (62) + 2 log (sqrt 6)
= 4 log 6 + log (sqrt 6)2
(x log y= log (yx) vice versa.)
= 4 + log 6

(log (in 6) of 6=1; any log (in x) of x=1)
= 4+1
= 5

Question 56

Write 3log(in 5)x-2log(in5)y3+log(in 5)7 as the logarithm of a single expression, assuming x and y are positive.

Solution 56

3 log x- 2 log y3 + log 7
=log x3 - log y6 + log 7
=log(in 5) (7x3/y6)
All the logs above are in 5.

Recall the following properties of logs:
x log y= log (yx)
log x - log y= log (x/y)
log x + log y= log (x*y)

Question 58

solve for X: 9^x/2+1=81

Solution 58

we know that 9^2=81 ( 9x9=81=>9^2=81)
=> 2= x/2+1
=> 2-1=x/2+1-1
=> 1=x/2
=> 1*2=x/2*2
=> 2=x


Question 59

How long does it take an $800. investment to earn $125 interest if it is invested at 6.5% interest compounded quarterly?

Solution 59

Comped Interest Formula:
If a principal p earns a quarterly yield of i(interest), then after n quarters there will be a total T, where
T=P(1+i)n (note: 1 quarter=3 months) so T=800+ 125=$925
i=6.5%=.065
P=$800

925=800(1+.065)n solve for n
=>925/800=(1+.065)n
=>ln(925/800)=nln(1+.065)
=>n=(ln(925/800))/(ln(1+.065))
=>n=2.3 quarters about 7 months


Question 60

What is matrix?

Solution 60

A matrix is a retaining array of numbers, matrixes provide a convenient shothand for working with systems of equations; for high school math, we need to know:

1.convert the systems of linear equations into a matrix form and the matrixes into the systems of linear equations;
2. identify the entries of rows and columns;
3. solve the systems of equations by using the Addition Method and similarly by row reduction of a matrix;
4. perform addition and multiplication on matrixes and vectors;
5.find the inverse to a square matrix( the numbers of rows and columns are the same);
6. compute the determinants of 2x2 and 3x3 matrixes;
7. compute the scalar(dot) product of two vectors.

Question 62

How to do multiplying & dividing decimal problem?

Solution 62

Let a= 123.456
a x 10= 123.456 x 10= 1234.56
a x 100= 123.45x 100= 12345.6
a x 1000=123.45 x 1000= 123456
When we multiply a decimal number with 10, 100, 1000.... we move the decimal point to the RIGHT one place, two places, three places....

a / 10 = 123.456 / 10= 12.3456
a/ 100= 123.456 / 100= 1.3456
a/ 1000= 123.456 / 1000= .13456
When we divide a decimal number by 10, 100, 1000....., we move the decimal point to the LEFT one place , two places, three places.....

If you move the decimal point to the right, the number will become bigger.
If you move the decimal point to the left, the number will become smaller.

Question 63

To approximate the speed of the current of a river, a circular paddle wheel with radius 4 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 10 revolutions per minute, what is the speed of the current? Express your answer in miles per hour.

t Solution 63

The speed of the paddle wheel = the speed of the current
1 revolution = the perimeter of the wheel= 2 (Pi) r
= 2 (pi) x 4 ft
=8 pi ft
10 rev/min= 8pix 10 ft/ min
= 80pi ft / min

1 mile= 880 fathom= 880x 2 yard= 880x2x 3 feet=880x6ft
1 hour= 60 minutes
So 80 pi ft/ min= (80pi/880x6 ) mile/ (1/60) hr
= 80 pi x60/880x6 mile/hr
= 10/11 mile/hr

Question 64

Naples, Florida, is approximately 90 miles due west of Ft. Lauderdale. How much sooner would a person in Ft. Lauderdale first see the rising sun than a person in Naples?

tSolution 64

1. 2 persons almost see the sunrise at the same time sine they are at the same time zone.
2. the speed of the rotation of the earth is 1 revolution per day.
1 revolution = the perimeter of the earth= 2 Pi r
r=6.4x103 km is the radius of the earth
=> 1 revolution= 2 x 3.14 x 6.4x103
=40192 km

Therefore the speed of the earth is:
t= (40192/ 24) km/hr=1675 km/hr= (1675/1.609) mile/3600 second= .2891 mile/sec

1 mile=1.609 km; 1 hour= 3600 second.
We are given that the distance between this two cities is 90 miles,so after T seconds the person on the west side will see the sunrise:
T= 90 miles / .2891 mile/sec = 311 seconds
=5.2 mintes

So the person on the east side will see the rising sun about 5.2 minutes sooner than the one on the west side.


Question 65

One number is missing from this list of perfect square between 100 and 300. give the missing number 121,144,169,196,256,289.

Solution 65

121= 11x11= 112; 144= 12 x 12 = 122; 139= 13 x 13 = 132; 196= 14 x 14 =142; so we miss 225= 15 x 15 = 152; 256 =16 x 16; 289= 17 x 17..

Question 66

Between which two consecutive integers is square62?

Solution 66

sqrt( 62) =~ 7.87, therefore sqrt( 62 ) is a number between 7 and 8.


Question 67

Evaluate sqt. 9*9+40*40.

Solution 67

(9*9+40*40)^2=1681^2=2825761


Question 68

Find the area of a square with sides of length square 2 c.m.

Solution 68

Area of a square= square os the length of the side= ( sqrt( 2) )2=2 cm2


Question 69

Find a counter example to this statement. the sum of two whole number is less then there product.

Solution 69

a=1, b= 2, 1+2=3 > 1*2=2


Question 70

How to convert square meters to square feet. For example, what is the equivalent of 200 sq. m. in sq. ft? Also, how do you convert square feet to square meters?

Solution 70

1 yard= 3 feet= 0.914 meter=> 1 foot= 0.914/3 meter= 0.305 meter and 1 meter= 3/ 0.914 feet=3.282 feet
Now we can square everything: 1 sq meter= (3.282)2 sq feet= 10.77 sq feet
1 sq feet= (0.305)2 sq meter= .093 sq feet
so 200 sq meter= 200x 10.77 sq feet=2154 sq feet;
200 sq feet = 200 x .093 sq feet= 18.6 sq feet.

Question 71

How long are the two trisectors of the rt. angle in a 3-4-5 rt. triangle?

Solution 71

Let's label the triangle as ABC counterclockwise from top to the bottom, where AB=4, BC=3, AC= 5. label the two trisectors( divide the right angle into 3 equal angle with degree 30) as BM and BN intersect AC with M and N.
Let the angle on the top to be A. so sin A = 3/5 and cos A= 4/5;
now let's use the law of sines to find BM and BN.
BM/ sin A= 4/ sin ( 180 --30 --A) => BM /( 3/5) = 4 / sin( 150 --A) =>
BM/ (3/5)= 4 / ( sin150cosA --cos150sinA)=> BM= 12/5 /[ (- 1/2)* 4/5--(sqrt3)/2 * 3/5)]
You can solve for BM by doing the algebra work;
BN/sin(90 --A) = 3/ sin ( 60 + A) => BN/ cos A= 3 / ( sin60cosA+cos60sinA)=>
BN= (4/5) (3/ [( sqrt 3)/2 * 4/5+ 1/2 * 3/5] ) you can do the algebra work from here for BN.

Remember the law of sines: for a triangle ABC
AB/ sinC= BC/ sin A= AC/ sin B

Question 72

Using {-1,2,3}as the domain for x, give the solution set of 7x-5(less then sign)2x.

Solution 72

x is in the domain of { -1, 2, 3}
7x-5 < 2x=> 5x - 5< 0=> 5x < 5 => x<1; therefore there is only one solution in the domain satisfy this unequality which is { -1} . check it yourself.

Question 73

solve y2=225

Solution 73

y2 = 225 => sqrt ( y2) = + and - sqrt( 225)= + and - 25. for 252= 225 and (-25)2=225


Question 74
Submitted on 10/23/1999

If s= the set of whole numberrs up to and including 10, how many member does s have.

Solution 74

s= { 0, 1,2 ,3 ,4, 5, 6, ,7 ,,8, 9 ,10} count the numberrs of the members in the set , we have 11 members in s


Question 75

Three instance of a pattern are given. describe the pattern by using one variable.
6+(6+2)+(6+3)=3.6+5
-2+(-2+2)+(-2+3)=3.(-2)+5
4/5(4/5+2)+(4/5+3)=3.4/5+5

Solution 75

can be expressed as:
a is any whole numberr :
a + (a+2) + ( a +3)= a*3 + 5


Question 76

A rectangle field is 160 feet long and 130 feet wide. to the nearest tenth of a foot,how long is diagonal of the field. [/color]

Solution 76

let L= length = 160 ft W= width = 130 D= diagonal
by using the pythagorean theorem , we have:
D2= L2 + W2 = (160)2 + (130)2 => D = sqrt ( ( 160)2 + ( 130)2)= 206.15 ft
We only consider the positive value here since it is a rectangle.


[b] Question 77

In the coordinate plane, given P(3, 2), Q(5, -1), the slope of PQ is ?

Solution 77

Slope is y2-y1/x2-x1 so -1-2/5-3=-3/2(slope of PQ)


Question 78

1) How many S atoms are in the 25.1g of Fe(subscript 2)S(subscript 3)?
2) How many P atoms are in the .0025 moles of Ca(subscript 3)(PO subscript 4)(subscript 2)?
3) If you have 1.82*10^91 atoms of Cl contained in a sample of AlCl(subscript 3), how much does this sample of AlCl(subscript 3) weigh??

Solution 78

1 mole of atoms = 6.022 x 1023 atoms
The unit of the molecular( or atomic) weight is g/mol: Fe: 56 g/mol; S: 32 g/mol; Ca: 40 g/mol; P: 30 g/mol; O: 16 g/mol; Al: 27 g/ mol; cl: 34.5 g/mol.(you can refer to the table in your textbook)

1.) the molecular weight of Fe(sub 2)S( sub 3) is 2 x 56 + 3 x 32= 208 g/mole
so 25.1 g will have 25.1 / 208= .12 mole of Fe(sub2)S(sub3)
1 mole of Fe(sub 2)S( sub 3) consists of 3 moles of S
so we have .12 x 3= .36 moles of S =>
there are .36 x 6.022 x 10^23 = 2.17 x1023 atoms of S in 25.1 g of the molecule.

2.) 1 mole of ca(sub3)(PO sub 4)( sub 2) consists of 2 moles of P.
We have .0025 mole of the molecule, which means there are 2 x .0025 = 0.005 moles of P in it.
Therefore we have 0.005 x 6.022 x 10^23= 3. 011 x 10^21 P atoms.

3.) First let's change atoms to moles for cl:
(1.82 * 1091)/ ( 6.022 * 10^23)= 3.022 x 1067 moles of cl and so we have 1 x 1067 moles for the whole molecule AlCl(sub 3), therefore the weight of it is:
(27 + 34.5 x 3) x 1 x 1067= 1.31 x 1069g


Question 79

A car travels at 80 kilometers per hour for 40 kilometers, 72 kilometers per hour for 48 kilometers, and then 96 kilometers per hour for 32 kilometers. What is the average rate of speed over the entire trip?

Solution 79

Let d1= 40km, s1= 80km/hr, need time t1 hr
so t1= d1/ s1= 40/ 80= 1/2 hr
Let d2= 48km s2= 72km/hr, need time t2 hr
t2= 48/72= 2/3 hr
Let d3= 32 km, s3= 96 km/hr need time t3
t3= d3/s3= 32/ 96= 1/3 hr
so we need to spend T= t1+t2+t3= 1/2 + 2/3 + 1/3= 3/6+ 4/6+ 2/6= 9/6= 3/2=1.5 hrs to run the total distance
D= d1 + d2+ d3= 40+ 48+ 32=130 km

Therefore the average speed is
S= D/ T= 130 / 1.5= 260/3 km/hr ( or approximately equal to 86.67 km/hr).


Question 80

I really don't understand rational, irrational, integers,natural. Is the numberr "pi" rational or irrational? Why? I have the definitions for rational ,irrational etc.

Solution 80

R= the set of real numberrs consists of all numberrs : include positive numberrs and negative numberrs, whole numberrs, zero, fractions and decimals themselves.
R: -5, 5, 0, 4, 1/2, -4/ 5, 3and1/3, sqrt(2).......

I = the set of integers include the whole numberrs and their opposite: -600, -4, -8/4=-2, 0, 1,2 ,3.......

N = the set of natural numberrs consists of only the positive integers: 1, 2, 3, 4, 5, 6.......

Rational numberrs is the set that can be written as a ratio of two integers: -2/3, -5/6....1/4, 5/6, 55/4, -5/1=-5, 3.5, 2.55555, -4.222222....(repeat with the same decimal 2) 4+1/2. ( integer, simple fraction, mixed numberrs, finite decimal or repeating decimal)

Irrational numberrs are that in the set of real numberrs, those numberrs are not natural numberrs . Every infinite decimal that does not repeat is an irrational numberr: Pi~ 3.141592653......( decimal doesn't repeat) sqrt(2)~ 1.4142135623.....

In fact any square root of any integer that is not an integer itself is an irrational number, sqrt(3) , sqrt(5) , sqrt(6), sqrt(10)....