Desmos for Geometry, Statistics, and others

Applying the same knowledge we covered in previously pages, we will do the same for these other topics of math

Geometry

$$x^2 + x + y^2 + y = \frac{199}{2}$$ The equation above defines a circle in the xy-plane. What is the length of the circle's radius?

Answer = 10

By plugging in the graph we will get this

  • We can select two points on the circle, across from each other, to determine the radius
  • Note: that the using the distance formula will only give us the diameter so divide that by 2 to determine the radius

$$x^2 +20x+ y^2 +16y = -20$$ The equation above defines a circle in the xy-plane. What are the coordinates of the center of the circle?

Answer = B

By plugging in the graph we will get this

  • We can select the points at top and the bottom of the circle to see that they both are at x=-10. Since the diameter from those two points will be a straight line the center must also must be at x=-10, which means B must be correct
  • We can further clarify this by using midpoint() function on desmos

What is the value of $$cos\frac{565pi}{6} $$

Answer = C

  • With questions like this we can plug into desmos and obtain a value of .86602 so make sure you check the value of the other possible answer choices

Right rectangular prism X is similar to right rectangular prism Y. The surface area of right rectangular prism X is 58 square centimeters ($$cm^2$$) and the surface area of right rectangular prism Y is 1450 square centimeters ($$cm^2$$). The volume of right rectangular prism Y is 1250 cubic centimeters ($$cm^3$$) What is the sum of the volumes, in ($$cm^3$$), of right rectangular prism X and right rectangular prism Y?

Answer = 1260

  • Similar shapes have a ratio of k. This k value scales in dimensionality as surface area is 2d,$$k^2$$,and volume is 3d, $$k^3$$. Using regression, we can quickly find the value of k, and then obtain the volume of Prism Y through the same method but with our K value.

In the xy-plane, the graph of the equation $$\left(x-3\right)^{2}+\left(y-5\right)^{2}=9$$ is a circle. The point(6,c),where c is a constant, lies on this circle. What is the value of c?

Answer = 5

  • First plug the circle equation into desmos.
  • We can use the fact that the point has as x value of 6 to create a vertical line at x=6, finding the intersection point and obtaining c

Side Note: Much of geometry and trigonometry is conceptual, so you wont need desmos most of the time unless its for arithmetic


Statistics

The dot plots represent 15 values in data set 1 and 15 values in data set 2. Which of the following statements about the standard deviations and ranges of the two data sets is true?

Answer = $$D$$

  • Although we can use the stdev function to make sure our answer is correct, for these types of questions, understanding it conceptually would be better as it takes time, meticulously plugging in each data point into each list.
  • Standard deviation measures the spread of the data, and we can clearly see the data for set 2 is centered heavily towards 12 while data set 1 is more spread out. Therefore, data set 1 will have a greater standard deviation as it is more spread out.

Data set A consists of 10 positive integers less than 60. The list shown gives 9 of the integers from data set A.

43,45,44,43,38,39,40,46,40

The mean of these 9 integers is 42.If the mean of data set A is an integer that is greater than 42, what is the value of the largest integer from data set A?

Answer = $$52$$

  • Notice how they ask for largest integer of the mean of data set A.
  • c acts as a changable 10th data point and we can see that when c=52 the mean of A, in this case b, is an integer. All other values of c less than 60 will result in a mean with a decimal

A data set of 27 different numbers has a mean of 33 and a median of 33. A new data set is created by adding 7 to each number in the original data set that is greater than the median and subtracting 7 from each number in the original data set that is less than the median. Which of the following measures does NOT have the same value in both the original and new data sets?

Answer = Standard deviation

  • We don't need to be restricted to the vague data set described above. Any data set with a reasonable quanitity of data points will provide the same result under the same modifications.
  • This is another question where understanding the problem conceptually is better than using desmos time-wise as it is obvious that the data will be more spread out, so standard deviation will change.

For a study, a research team measured the heights of 380 king penguins from the Kerguelen Islands and 190 king penguins from the Falkland islands. The research team determined that the mean height of 380 king penguins from the Kerguelen Islands was 87 centimeters and the mean height of the 190 king penguins from the Falkland Islands was 93 centimeters. What was the mean height, in centemeters, of all 570 king penguins the research team measured for this study

Answer = $$89$$

  • We can use regression to quicken the proccess
  • Know that mean is the sum of the data points divided by the total amount of data points. We were finding the sum in this case.

Side Note: Although desmos can be helpful on the statistics section most of time understanding the problem conceptually will be faster.


Percentages

The number a is 190% greater than the number b.The number b 80% less than 24.What is the value of a?

Answer = 13.92

  • 190% greater means add an extra 190 to the 100%.
  • 80% less means subtract 80 from 100%
  • Now we can set up the regression, using the % to remove the need of converting 290% and 20% into 2.9 and .2

In 2008, Zinah earned 14% more than in 2007, and in 2009 Zinah earned 4% more than in 2008. If Zinah earned y times as much in 2009 as in 2007, What is the value of y?

Answer = D

  • If say the money made in 2007 is "a" and set it to 1, we can set up the regression as every part will be assigned to some number.
  • Money made in 2008 is b = 1.14a, and money made in 2009 is c = 1.04b. We can see that every element in the list has a value assigned to it and if we take the ratio of c/a we can find y
    We derived the ratio from c = 1.04b -> c= 1.04(1.14a) -> $$\frac{c}{a}$$ = 1.1856