Distance, Midpoint, and Slope

This week’s topic has been Distance, Midpoint, and Slope.

Let’s go through that blue SAT book everyone has, http://amzn.to/1wADKLS, or http://amzn.to/1wtPZuD, and this ACT book for problems about distance, midpoint, and slope that can be solved using my TI-84 Plus programs DISTANCE. Sorry, I won’t post the actual problems here.  But go dig out your books and calculator (buy them here: SAT, ACT , TI-84 Plus) and follow along.

SAT book

ISBN-13: 9780874478525.  The page numbers included here are from that book, but if you have the other one (9780874479799), please refer to the Test number and Section number instead.

DISTANCE:

Test 1, Section 3, p. 397, problem # 6

Test 1, Section 3, p. 400, problem #15

Test 3, Section 8, p. 546, problem #10

Test 4, Section 6, p. 596, problem #10

Test 5, Section 8, p. 669, problem #8

Test 7, Section 3, p. 769, problem #4

Test 8, Section 3, p. 830, problem #2

Test 9, Section 5, p. 905, problem #8

Test 10, Section 2, p. 953, problem #17  

ACT book

I’m working out of an older edition that only has 3 tests.  I believe they are the same as the first 3 tests of the 2nd and 3rd editions.  Once I have the newer book, I’ll add to this list.

DISTANCE:

Test 1, Section 2, problems #35, 45

Test 2, Section 2, problem #33

Test 3, Section 2, problems #38, 48  

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Slope

This week’s topic is:  Distance, Midpoint, and Slope.

Q:  The slope of the line passing through (-1,-3) and (7,y) is -1/2.  What is the value of y?

.

.

.

.

Answer: -7

Algebraic solution by hand:

Use the slope formula:

 

2014-11-07_153414

Solution using my TI-84 Plus program DISTANCE:

Run DISTANCEHow?

Use your multiple choice answer choices.  One of them is -7.  (If you don’t have choices, draw a picture.  Plot the point (-1,-3) and do a slope of -1/2 from there – down 1, right 2, down 1, right 2, … until you get to a point where x is 7.  Read the y-value of that point – it will be -7 and check it using this program.)

x1=-1

y1=-3

x2=7

y2=-7

Ignore the distance information – we don’t need it for this problem.  Press ENTER for the midpoint information, which we don’t need either.  Press ENTER once more to find that the slope is -1/2.

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Distance

This week’s topic is:  Distance, Midpoint, and Slope.

Q:  For what value(s) of x is the point (x,4) exactly 5 units away from the point (6,8)?

.

.

.

.

Answer: 3 or 9

Solution by hand:

Use the distance formula:

distance

Solution using my TI-84 Plus program DISTANCE:

Draw a picture:

distgraph

Guess that x=2, maybe.  It’s probably an integer if this is an SAT problem.  Use the answer choices if available.

Run DISTANCEHow?

x1=2

y1=4

x2=6

y2=8

The distance is 5.66.  It’s too big – you wanted it to be 5 – so move the point a little closer to the middle.

Maybe x=3:

x1=3

y1=4

x2=6

y2=8

The distance is exactly 5.  Perfect.  (There’s also another possible answer, x=9.  If you have multiple choice answers, you will see 3 and 9 as a choice, so try both and see that they both work.  For a grid-in SAT problem, you would only have to find one answer anyway.)           :

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Ratios

Q: Last month, the ratio of sunny days to rainy days was 3:2.  How many rainy days  were there last month, if the month had 30 days?  (Assume all days were either sunny or rainy.)

 

.

.

.

.

Answer: 12

Solution by hand:

Number of sunny days = 3x

Number of rainy days = 2x

3x + 2x = 30

5x=30

x = 6

There were 2x rainy days, so there were 12 rainy days.

 

Solution using my TI-84 Plus Program RATIO:

Run RATIOHow?

Total number of items (days) = 30

Number of categories of items = 2  (sunny and rainy)

Ratio part for category 1 (sunny) = 3

Ratio part for category 2 (rainy) = 2

Distribution is: sunny = 18 and rainy = 12.

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Proportions

Q:  x is inversely proportional to w.  If x = 6 when w = 8, what is x when w = 12?

.

.

.

.

 

Answer: 4

Solution by hand:

For inverse proportions, (x1)(w1) = (x2)(w2)

(6)(8) = (x)(12)

48 = 12x

x=4

Solution using my TI-84 Plus Program PROPORTN:

Run PROPORTN.  How?

x1 = 6

y1 = 8  (here, we have renamed w1 as y1)

Have y2 (really w2)

y2 = 12

Choose ‘Inverse’ proportion.

Then, x2 = 4.

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Proportions

Let’s change yesterday’s question around a bit:

Q:  Wandering around blindly in a parking lot (don’t try it, please), the probability that the first car you bump into is blue is 2/5.  If there are 500 blue cars, how many cars are in the parking lot?

.

.

.

.

 

Answer: 1250

Solution by hand:

Set up the proportion using the concept ‘blue / total = blue / total’:

2/5 = 500/x

2x = 2500

x = 1250

Solution using my TI-84 Plus Program PROPORTN:

Run PROPORTN.  How?

x1 = 2  (blue)

y1 = 5  (total)

Have x2 (blue)

x2 = 500

Choose ‘Direct’ proportion (because it doesn’t say ‘inverse’ in the problem).

Then, y2 (the total) = 1250.

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Ratios

Q:  The ratio of blue cars to non-blue cars in the parking lot is 2:7.  If there are 1260 cars in the parking lot, how many are blue?

.

.

.

.

 

Answer: 280

Solution by hand:

Number of blue cars = 2x

Number of non-blue cars = 7x

2x + 7x = 1260

9x=1260

x = 140

There are 2x blue cars, so there are 280 blue cars.

 

Solution using my TI-84 Plus Program RATIO:

Run RATIOHow?

Total number of items = 1260

Number of categories of items = 2  (blue and non-blue)

Ratio part for category 1 (blue) = 2

Ratio part for category 2 (non-blue) = 7

Distribution is: blue = 280 and non-blue = 980.

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

SAT – Probability

Q:  What is the probability of randomly guessing the last four digits of a telephone number correctly?

.

.

.

.

scroll down for answer

.

.

.

.

 

 

 

A: 1/10,000

Explanation:  Each digit is chosen from the integers 0 through 9, which is ten possible choices for each digit.  There are four digits to choose independently, so there are:

10 x 10 x 10 x 10 = 10,000 ways to choose the four digits.

Then, the one correct sequence of 4 digits, is one out of 10,000.  The probability is 1/10,000.

Another way to arrive at the number 10,000 is as follows:

The 4-digit sequence may be anything from 0000 to 9999.  So, how many is that?  9999 – 0000 +1 = 10,000.  Why the ‘plus one’ on the end?  If you just subtract the numbers, it tells you how many spaces there are between the numbers.  Add one to include either the first number or the last number, whichever one has not been already counted.

 

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Precalculus – Domain of a function

Q: What is the domain of this function?

 

Answer:

   ‘Like’ if you got it!

Explanation:

The domain of a function is a description of the x-values that may be plugged into the function. For a square root function, the restriction is that the number under the radical must not be negative. Therefore,

 

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

ACT , Algebra 1 – system of equations

MathPro Q&A Forum (ACT)

 Q: For what value of A will the following system of equations have infinitely many solutions?

3x – 2y = 4

Ax + 4y = -8

 

Answer: -6

Explanation: In order for the system to have infinitely many solutions, the two equations must describe the same line.  (Recall that the solution(s) of a system are the points where the graphs of the two lines meet.  For the two lines to meet in infinitely many points, they must be the same line.)

 To make the second equation the same as the first, multiply both sides of the first equation by -2:

 -2(3x – 2y) = -2(4)

-6x + 4y = -8

So, A = -6.

Highly Effective Headline!
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.