6 Geometry - MC
A staircase consists of four steps and four risers. The step at the upper left if 1.5 ft long. Each of the other steps is 12 inches long. Each of the four risers is 9" high. The staircase is 2 yds wide. How many square ft of carpeting is needed to cover the steps & risers? How many square inches and sq. yards are needed? How many cubic ft of concrete is needed to build this? [12" = 1 ft // 1 yd = 3 ft // 1ft^2 = 144"// Rec prism - Vol = l w h.]
(12"= 1ft)// (1yd= 3ft)// 9"= 9/12 = .75ft // 2yd = 2(3ft) = 6ft // (6 x 1.5) + 3 (6 x 1) + 4 (6 x .75) = 45 ft^2 // 1ft^2 = 144" = (12 x 12)// 45 x 144 = 6,480 in^2 // (3x3)= 9 // 45/9 = 5 yd^2 // Rec prism - Volume: l w h // RP#1 - (1.5)(6)(3) = 27 ft^3// RP#2 - (1)(6)(2.25) = 13.5ft^3// RP#3 - 9ft^3// RP#4 - 4.5ft^3 = 54ft^3
Fractions to decimals - A number line shows C (4/6) and D (5/6). This creates a value that lies between C and D on the number line ____
1/3 + 2/5 = 11/15 \\ Use decimal approxim. 4/6 = 0.66 and 5/6 = 0.83. C = 11/15 = 0.73 - between (0.66 and 0.83)
Division - If your electric vehicle has a 0% charge and you want to charge it to 100% capacity, how long will it take a standard 12 -amp (120-volt) outlet to charge your EV that has a battery capacity of 16,000 watts? 12amp/120volts= 1,440 watts/hour.
16,000 / 1,440 watts = 11.11 hours
Square feet - How many square feet are there in 2 square yards?
18 sq ft = (3x3) +(3x3) // 3ft = 1 yd // (3 x 3) = 9 sq. ft = 1 sq yd
Triangle ABC is isosceles. Angle B = 120 degrees. What is the measure of Angle A?
180-120 = 60 || 60/2 = 30 degrees. Isosceles - triang. two sides of same length. Interior angles of triangle add up to 180 deg. acute < 90, obtuse > 90
Multiply - A plumber charges d - dollars per minute plus $25.00 for driving time. What is the total amount charged for a job that takes 3 hours?
180d + 25 // 180 min = 3 hours
Place value - What value does the 2 represent in the number 2.1 x 10^ -3?
2/ 1,000 // -3 = 000
There are 300 marbles in a collection: green (30%), yellow (10%), blue (20%). The rest are red or black. Find the percent of the marbles that are either red or black. If 30% of the marbles that are either red/back are red, how many are black? If a marble is randomly selected, the probability that it is black is ___.
30% of 300 = 90 (green) // 10% of 300 = 30 (yellow) // 20% of 300 = 60 (blue) // 30+10+20 = 60%. // 100-60 = 40% (red or black) // 40% of 300 = 120 (red/black). 30% of 120 = red. 70% of 120 = 84 (black). 84/300 = 7/25 (28%) probability that it is black.
Parallelogram - A (2,2) , B (6,2), C (x,y), D (4,4). The coordinates of point C are ___
4 + 4 = 8 // (8, 4)
Oranges in a supermarket are stacked in the shape of a pyramid. Each layer of the stack is a square. The bottom layer is a square with 5 oranges on each side. The side of each successive layer is made shorter by one orange. The total number of oranges that can be stacked is ___.
5 x 5 + 4 x 4 + 3 x 3 + 2 x 2 + 1 = 55 oranges
Find the product of (x + 5) and (x + 3). Demonstrate and explain using an algebraic method and a geometric method.
Algebra. - multiply the expressions using the distrib. property. The x in the first parentheses is distrib. over the (x+3) and the "5" in the first parenth. is distrib. over the (x+3). Then simplify the combining like terms. (x+5) (x+3) = x(x+3) + 5(x+3) = x^2 +8x + 15// Geom. - create a rectangle with sides - length: (x+5) and width (x+3). Find the area of the rectangle. Inside each of the smaller rectang. is a value that represents its area. The area of the larger rect. can be expressed as the sum of the smaller rect inside it.
Shaded area - Circle O is inscribed in square ABCD. AB= 10. The area of the shaded region [one corner of square - circle] is closest to the following choice. O - A = T r^2
Area of Square = 10 x 10 = 100 // Area of Circle = T r^2 = T (5)^2 = 3(5)^2 = 75 // 100 - 75 = 25 // 25 /4 = 6.25
Shaded area - A large square consists of squares and isosceles right triangles. Four small squares inside are UNshaded. The remainder of the objects are: 1 shaded square and 12 triangles. If the large square has sides of 4cm, the area of the shaded portion in sq cm is __
Area of large square = 4 x 4 = 16. Group shaded shapes - 4 squares. Unshaded squares = 4 // 16/2 = 8 cm^2
A triangle ^ ABC contains a smaller ^ ARS inside of it. To find the length of a lake, surveyors measure the distances shown such that ^ ABC and ^ ARS are similar. If RS = 3km, AS = 4km, and SC = 2km, what is BC, the length of the lake?
BC / RS = AC/ AS || BC / 3 = 6 /4 || BC = 18 /4 = 4.5 km
The point of a compass is placed on point A of line segment AB and an arc was drawn. POC was placed on point B of line segment AB, and another arc was drawn. The intersections of the arcs were connected, forming line segment CD.
CD is perpendicular to and bisects AB
A right circular cylinder RCC has a base diameter of 4 inches and a height of 7 inches. Find the surface area and the volume and explain the steps used. O - A = Tr^2 // Rect - A = (l) (w) // Volume of RCC = Tr^2 x h
Diameter - 4 ", Radius - 2 " // O - A = Tr^2 = T(2)(2) = 4T = Area // Top O + bottom O = 8 T // Rect - A= length x width. 4 T x 7 = 28T // 28T + 8T = 36 T square inches = surface area. // Vol of RCC = Tr^2 x h = T(2)(2) x 7" = 28 T cubic inches
Four congruent triangles, each having legs of length a and b and hypotenuse of length c, are arranged to make square EFGH. Write an expression for the area of square EFGH in terms of the lengths of its sides, its component parts (4 triangles + square) and set these two expressions equal - proving the Pythagorean theorem (a^2 + b^2 = c^2)
EFGH = (a+b) ^2 // (a+b)^2 = 4 (1/2 a x b ) + c^2 // a^2+ 2ab +b^2 = 2ab + c^2 // a ^2 + b^2 = c^2 - Pythag theorem. Congruent - two objects with the same shape and size - can be repositioned/ reflected. hypotenuse - longest side of a triangle, opposite right angle
Create a square wall from RECtangular tiles that must be twice as long as they are wide. Wall is 60 inches square. All rec. tiles must have measure. that are in whole numbers of inches. No overlapping tlies, empty spaces, or tiles going beyond the borders. Find three diff. sizes of these rec. tiles and find how many of each size could be used. Find the fewest # of rect. tiles that could fill the wall.
Factor 60. 60 = 12 x 5 = 2 x 2 x 3 x 5. (1 x2) , (2x4), (3x6) // (1x2) - 60/2 = 30 // 30 x 60 = 1800 tiles // (2x4) - 60/4 = 15// 15 x 30 = 450 tiles // (3 x 6) - 60/6 = 10 // 10 x 20 = 200 tiles. Chart - l = 2w. (2,1) (4, 2) (6,3) ... (30/15), (60/30) // 60/30 = 2 fewest rect tiles.
An artist is constructing a rectangular wall design from square tiles. The wall is 72" long and 42" wide. All the sq. tiles must be the same size, and the length of the sides of the tiles must be a whole number. Find three different sizes of sq. tiles that can be used to fill the space, with no tiles overhanging the border. Find the smallest number of sq. tiles that can be used to fill this space.
Factor 72 and 42 to find = 2 x 2, 3 x 3, 6 x 6" sq. tiles || (6 x 6) largest tile // 42 / 6 = 7 and 72 / 6 = 12 || 12 x 7 = 84 tiles - the smallest # of sq. tiles needed to fill the space
x^2 + 3x + 2 = 0 is a quadratic equation. Solve this equation and explain the steps used.
Factor the first term, x^2 = ( x ) ( x ) // Factor the last term, 2. (2 x 1) = (x + 2) (x + 1) // multiply the means/extremes together = 3x. trinomial is properly factored. set each factor equal to 0 // (x + 2) (x + 1) = 0 // x = -2 or -1
Three circles are divided and shaded in grey: 5/8, 1/6, 3/7. What is the sum of the shaded areas if each circle represents one unit?
Find common denom. 8 x 6 x 7 = 336 // add numerat. 210 + 56 + 144 = 410/336 = 205/168 = 1 37/168
Hypotenuse - Point P (1,1) and Q (1,0) lie on the same coordinate graph, the following must be true ___
P is farther from the origin than P is from Q // Triangle PQO form a right triangle with hypotenuse OP. Hypot. is the longest side of a right triangle (oppos. right angle).
If point P (1,1) and point Q (1,0) lie on the same coordinate graph, the following is true:
P is farther from the origin than P is from Q. Hypotenuse OP is longer than PQ
Probab. - Two dice with number rep. from 1-6 are tossed. What is the prob. that the product of the two numbers shown is less than 20?
Prob. = #favored/ #total // Table - (1-6) x plane and (1-6) y plane. 36 total spaces - possible outcomes. Checks placed in squares where the product is less than 20. Product = multiply. = 28/36 = 7/9
Figure A: (-1, 4), (3, 4), (3, 2). Figure B is the result of reflecting Figure A over the y-axis. Rotate the figure 90 degrees clockwise. Which of the following are the coordinates for Figure B (congruent)?
Reflect the y-axis: (1, 4), (-3, 4), (-3, 2) // multiply x and (-1) // 90 degree clockwise rotation : (4, -1), (4, 3), (2, 3) x and y rotate // multiply y and (-1)
Pythagorean Theorem
a²+b²=c² // finds the distance between two points given their coordinates
Point E is on a line with a slope of 2 in the x-y plane. E = (2 , -1). Which of the other points is also on the line? Use the graph to answer the Q.
slope (m) = 2, 1 // 2^, 1 > // B = (4, 3) is also on the line
A rectangle is 8 cm long, 3 cm wide, and 4 cm high.
the surface area is: 136 cm^2 // 24 x2 = 48 , 32 x2 = 64, 12 x 2 = 24// 48 + 64 + 24 = 136 cm^2
Upward curve - A table with coordinates (1,1), (2,4), (3,9), (4,16) - The following graph best represents the data:
x^2 = y. When x is 3, y is 9. The graph with the upward curve is the correct one
