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GAT Final Q3
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Gravity
Terms in this set (145)
What is the Lateral Surface Area(LSA) of a prism?
The sum of the areas of all the lateral faces.
What is the Total Surface Area(TSA) of a prism?
LSA+2(Area of all the bases)
What is the Volume of a prism?
(Area of the base)(Height of the prism)
What is the Lateral Surface Area(LSA) of a pyramid?
Add the areas of all the lateral triangular faces.
What is the Total Surface Area(TSA) of a pyramid?
LSA + Area of the base
How do you find the volume of a pyramid?
1/3(Area of the base)(Height of the pyramid)
What is the vertex of a pyramid?
Point not in the same place as the base of the pyramid.
What is the slant height of a regular pyramid?
The length of the altitude of any lateral face from the vertex of a pyramid
What is a regular pyramid?
A regular pyramid is a pyramid that has a regular polygon for a base and lateral edges that are congruent.
What is the base of a pyramid?
It is a polygonal region determined by all but one vertex of a polyhedron.
What shape is a lateral face of a pyramid always?
A triangle.
What is the height of a pyramid?
The length of the altitude of a pyramid.
What is a lateral edge?
A side of a prism/pyramid.
What is a lateral face?
A face of a pyramid/prism that isn't the base.
What is a vertice?
A point on a prism/pyramid.
What are the five platonic solids?
1.) Tetrahedron
2.) Cube
3.) Octahedron
4.) Dodecahedron
5.) Icosahedron
What type of face does each platonic solid have?
1.) Tetrahedron- triangle
2.) Cube- square
3.) Octahedron- triangle
4.) Dodecahedron- pentagon
5.) Icosahedron- triangle
What are the duals of the platonic solids?
1.) Tetrahedron - Tetrahedron
2.) Cube - Octahedron
3.) Octahedron - Cube
4.) Dodecahedron - Icosahedron
5.) Icosahedron - Dodecahedron
average(statistics)
Any one of several measures designed to reveal the central tendency of a collection of data.
Categorical data
Data that can be sorted into different categories distinguished by a non numeric characteristic.
Continuous Data
Data resulting from infinitely many possible values that can be associated with points on a continuous scale in such a way that there are no gaps or interruptions
Descriptive Statistics
Methods used to summarize the key characteristics of known population data
Five-number summary
Minimum score, maximum score, median, lower quartile, upper quartile
Interquartile Range(IQR)
Difference between the third and first quartiles
Lower Quartile(Q1)
Median of the lower half of all scores (from the minimum score up to and including the original median); used in a boxplot
Mean
Sum of a set of scores divided by the number of scores
Median
Middle value of a set of scores arranged in order of magnitude (value)
Mode
Score that occurs most frequently
Population
Complete and entire collection of elements to be studied
Quartiles
The three values that divide ranked data in four groups, with approximately 25% of the scores in each group
Random Sample
Sample selected in such a way that every member of the population has the same chance of being chosen
Range
Measure of dispersion that is the difference between the highest and lowest scores.
Sample
Subset of a population
Upper Quartile(Q3)
Median of the upper half of all scores (from the original median up to the maximum score); used in a boxplot
Bimodal curve
Two peaks (of approximately) the same height.
Distribution
Refers to the way the data is spread out on the number line.
Gaps
Intervals where there are no data between the low and high.
Outlier
A value that is far from the rest of the data
Variability
Refers to how much the data is spread out on the number line
What is the test for outliers?
-1.5 times the interquartile range + third quartile
-First quartile - 1.5 times the interquartile range
Example:
Q1 = 5
Q3 = 8
IQR = 3
1.5 times 3 = 4.5
4.5 + 8 = 12.5
5 - 4.5 = 0.5
Any data point less than 0.5 or greater than 12.5
is an outlier. In this set, 15 is an outlier.
Line Plot
Possible data values are listed on the x axis. An X goes for every element of the corresponding value.
Stem Plot(stem and leaf plot).
Plot that seperates data by 10's. Stems represent each 10 value and the leaves are the 1 value.
Histogram
a diagram consisting of rectangles whose area is proportional to the frequency of a variable and whose width is equal to the class interval.
Box Plot(box and whiskers plot)
Constructed by marking the 5 point summary
How do you find standard deviation?
What percentage of data is found within the first standard deviation?
68%
What percentage of data is found within the second standard deviation?
95%
What percentage of data is found within the third standard deviation?
99.7%
What percentage of data is found within the fourth standard deviation?
100%
In a DOE, what is the range of standards and why is it important?
The range of standards is the highest standard minus the lowest standard. You need this to find out what is statistically significant.
How do you find out if something is statistically significant?
2(range of standards)
Data within this number is not significant.
How do you find the effect of a variable in a DOE?
You subtract the average of the (-) values from the average of the (+) values.
How do you find the interaction effect in a DOE?
You subtract the slopes.
Ex.) Time over Heater
(-) (+)
Pennies (+) 6 13
(-) 29 39
So 13-6/1-(-1) - 39-29/1-(-1) =
3.5-5 = -1.5
-1.5 is the interaction effect.
What is the prediction equation?
Y= grand average + effect of predictor value1/2
variable1 + effect of predictor variable2/2
variable2 + effect of the interaction/2
variable1
variable2+ "noise"
What is the parsimonious equation?
Y = grand average + statistically significant values/2 + "noise"
What is a circle?
A set of points on a plane at a distance r(radius) from a point P(center)
What is a chord?
Line segment whose end points are on the circle.
What is a secant?
A line that intersects the circle in 2 distinct points.
What is a tangent?
A line in the plane of a circle that intersects the circle at one point.
The radius of a circle is ______ to the tangent line at the point of tangency.
Perpendicular.
What is a sphere?
Set of all points in space whose distance from the center is a distance r(radius)
What are concentric circles?
Coplanar circles with the same center but different radii.
What is the equation of a circle?
x²+y²=r²
How do you find the square of a circle?
y= ax²+bx+c
A. Make sure the value of a is monic(=1)
B. Add plus blank(c) to both sides
C. Take 1/2 of b, square it, and add the number to both sides of the equation.
D. Factor into (x+1/2b)²
Find the square of x²+14x=23
(x+7)²= 72
Write the equation of a circle where (0,0) is the center and 5 is the radius.
x²+y²= 25
Translate the circle with a center of (0,0) and radius of 5, 4 units to the right and 3 units down.
(x-4)²+(y+3)²=25
Perpendicular lines have ________________ slope.
Opposite reciprocal slope.
ex.) the perpendicular slope of 2/9 is -9/2
What is the midpoint formula of a 2D object?
[(x_1 + x_2)/2 , (y_1 + y_2)/2]
What is the distance formula of a 2D object?
d =square root of (x2 - x1)² + (y2-y1)²
What is the name for the point of concurrency for the Angle Bisectors of a Triangle? Think of your acronyms.
Incenter
What is the name for the point of concurrency for the medians of a triangle? Think of your acronyms.
Centroid.
What is the name for the point of concurrency for the altitudes of a triangle. Think of your acronyms.
Orthocenter
What is the name for the point of concurrency for the perpendicular bisectors of a triangle? Think of your acronyms.
Circumcenter
The incenter of a triangle is equidistant from the _____ of the triangle
Sides
The circumcenter of a triangle is equidistant from the ________ of the triangle.
Vertices.
The centroid is _____ of the distance from each vertex to the midpoint of the opposite side.
2/3
To inscribe a circle about a triangle, you use the _________.
Incenter
To circumscribe a circle about a triangle, you use the ___________.
Orthocenter.
The point of concurrency for the incenter and centroid is always _______ the triangle.
inside.
The point of concurrency for the circumcenter and orthocenter for each angle(acute, obtuse,right) is....
Acute- inside the triangle
Obtuse- outside the triangle
Right- on the triangle
How do you find the y and x intercepts of a circle.
You plug in 0 into the circle equation.
Know how to 3D graph.
...
What is the midpoint formula for 3D objects?
What is the distance formula for 3D objects?
What's the formula for the volume of a cone?
Volume = 1/3πr² · h(height)
What's the formula for the Surface Area of a cone?
πr² + πr · l(slant height)
What's the formula for the volume of a cylinder?
πr²·h
What's the formula for the surface area of a cylinder?
2πr² + 2πr·h
or
2πr² + π·d(diameter) · h
What's the formula for the volume of a sphere?
4/3πr³
What's the formula for the surface area of a sphere?
4(πr²)
How do you find the volume of a frustrum(bucket shaped object)?
1. Extend the frustum to make it a cone. Then you take the ratios of the height and set it equal to the ratios of the bases, solve to find height, and subtract the volume of the small come from the big one
or
2. πh/3(R²+Rr+r²)
What is a minor arc?
An arc that has less than 180°
What is a major arc?
An arc that has more than 180° but less than 360°
What is a semicircle?
A semicircle is equal to 180°
What is a central angle?
A central angle has its vertex at the center of the circle and its rays intersect the circle. A central angle is equal in measure to its intercepted arc.
(in picture angle aoc is equal to the measurement of arc ac)
What is an inscribed angle?
An inscribed angle has its vertex on the circle and its rays intersect the circle. An inscribed angle is half the measure of its intercepted arc.
How do you find the length of an arc?
2πr • central angle/360
How do you find the area of a sector?
Central angle/360·r²π
How do you find the circumference of a circle?
c = πd
d = diameter
If the measurement of a central angle is 38 degrees, then the measurement of the arc is...?
38 degrees
If the measurement of an inscribed angle is 24 degrees, then the measurement of the arc is...?
48 degrees
When finding the surface area of a cone, what represents the central angle?
Ø
(slash should be straight)
When finding the surface area of a cone, how do you find the length of the arc(circumference of the smaller circle?
2πr
When finding the surface area of a cone, how do you find the circumference of the new circle?
2πr · 360/Ø
When finding the surface area of a cone, how do you find the diameter of the new circle?
(2πr · 360/Ø)/π
When finding the surface area of a cone, how do you find the radius of the new circle?
r·360/Ø
r = radius of small circle
Or divide diameter by 2
When finding the surface area of a cone, how do you find the area of the sector of the new circle?
πr²· Ø/360
r = radius of the larger circle.
If the ratio of the radii of two spherical regions is 3 to 1, what is the ratio of their areas?
9:1
If the ratio of the radii of two spherical regions is 3 to 1, what is the ratio of their volumes?
27:1
If the ratio of the surface areas of two spherical regions is 16 to 1, what is the ratio of their radii?
4:1
Do the rows or columns come first?
Rows
What is a matrix?
A matrix is a rectangular arrangement of objects, each of which is called an element of the matrix.
A point matrix is always written as a _ x _ matrix.
2 x 1
Can you add a 1 x 3 matrix with a 3 x 1 matrix?
No.
Can you add a 1 x 3 matrix with a 1 x 3 matrix?
Yes.
How do you scale the matrix [2 3 4] by 3? What is the final matrix look like?
3[2 3 4]
[6 9 12]
What's the easiest way to transpose your matrix?
Use your calculator.
How do you multiply matrices?
You find the sum of the products of corresponding elements from the ROWS of the first matrix and the COLUMNS of the second matrix.
Can you multiply a (1x 3) matrix with a (3 x1) matrix?
Yes because the inner dimensions match up.
Can you multiply a (4 x 2) matrix with a (1 x 5) matrix?
No because the inner dimensions don't match up.
What are the dimensions of the product of a (2 x 2) matrix and a (2 x 1) matrix?
(2 x 1)
What are the dimensions of the product of a (3 x 4) matrix with a (5 x 6) matrix?
There isn't one because the inner dimensions don't add up.
When using matrices for data, the only numbers that mean anything are the ones along the ________.
Main diagonal.
The Element through some operation with the identity produces the _________.
Original element
The element through some operation with the inverse produces the ________.
Identity.
The additive identity of 5 is...?
0 because 5 + 0 = 5
5 = original element
The additive inverse of 5 is...?
-5 because 5 + -5 = 0
0 = identity
The multiplicative identity of 5 is...?
1 because 5 · 1 = 5
5 = original element
The multiplicative inverse of 5 is...?
1/5 because 5 · 1/5 = 1
1 = identity
The identity of a 2x2 matrix is...?
1 0
0 1
How do you find the inverse of a 2x2 matrix?
1. Find Determinant(det)
det=A·D - B·C
2. Divide each element by determinant.
3. Swap the position of A and D. Multiply B and C by -1
How do you find the inverse of a larger matrix?
Put that matrix in your in-spire calculator and make it to the -1 power.
What are the 3 steps to solving a systems of equations with matrices?
1. A · [x y](actually 2x1) = B
2. A^1·A[x y](2x1) = A^-1 · B
3. 1 0 x
0 1 · y = A^-1· B
Compare and contrast the volume of a triangular prism with the volume of a triangular pyramid.
The volume of a triangular prism is Area(base)·height. In triangular prisms, the area of the base is 1/2bh, so the actual formula is (1/2bh)(height). The volume of a triangular pyramid is 1/3Areabase·height. Just like the triangular prism, the bases are triangular so 1/2bh can be substituted in for the area of the base. The formula for the volume is 1/3(1/2bh)(height). It is 1/3 because 3 pyramids can fit into 1 prism.
Compare and contrast the mean versus the median in a data set.
The mean and the mode both show a central amount. They are both around the middle. The mode is the sum of the amounts divided by the number of data elements. The median is the central amount. Both numbers are usually close to each other but not always exact.
Compare and contrast the secant of a circle with a tangent of a circle.
The secant of a circle is a line that goes through and intersects the circle twice and never ends. A tangent is a line outside the circle but it intersects once. This tangent is perpendicular to the radius of the circle. Both lines never stop and intersect with the circle at least once.
Compare and contrast the incenter of a triangle with the circumcenter.
The incenter of a triangle is created by angle bisectors. The lines of the incenter always meet inside the circle. They also come from the angles of the triangles. The circumcenter is created from perpendicular bisectors. The area in which they intersect depends on the angle measures of the triangle. They come from the lines of a triangle. They have points of concurrency and intersect inside the triangle with the angle measures are acute.
Compare and contrast the orthocenter and circumcenter of a triangle.
The orthocenter and circumcenter are both points of concurrency. When the angles are obtuse, the point of concurrency is outside the triangle, and inside for acute. The point of concurrency is on the triangle when there's a right triangle though. These are different than the incenter and centroid, which all intersect regardless of angle measure. The orthocenter is created from the altitudes of the triangle and the circumcenter is created by perpendicular bisectors, both which create 90°.
Compare and contrast identity matrices with inverse matrices.
The identity of a matrix is the matrix when gone under some operation, produces the same matrix. The inverse of a matrix is the matrix when gone under some operation produces the inverse. The inverse matrix of 2x2 matrix is
1 0
0 1 . So if any matrix is multiplied by that matrix equals the original matrix.
True or False: If a diametral chord of a circle bisects a chord that is not a diameter, then the diamatral chord is perpendicular to the other chord.
True
True or False: Any angle inscribed in a semicircle is a right angle.
True
True or False: IF the interiors of two spheres both contain a given point, then the intersection of the spheres is a circle.
False
True or False: If two arcs of different circles have the same measure, then their chords are congruent.
False
Give a three-dimensional open sentence that describes the set of all points that are 8 unts fro, the point (1,-1,2).
(x-1)²+(y+1)²+(z-2)²= 64
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