Out of 60 review questions I struggled with these ones. Someone please help!!
1.) The length of one side of a regular polygon is half the length of a similar polygon. What is the ratio of their areas?
a.)1/2
b.)1/4
c.)1
2.)A patchwork quilt is made up of two different sized squares. The length of the larger square is twice the length of the smaller square. How many of the smaller squares does it take to equal the area of one larger square?
a.)2
b.)4
c.)8
3.)In any prism, the lateral edges are congruent and parallel.
a.)Sometimes true
b.)always true
c.)Never true
4.)A measuring cup is shaped like a cylinder and has a radius of 5 cm. It is filled with milk to a height of 2 cm. What is the lateral area of the milk in the measuring cup?
a.)20Ï€ cm^2
b.)10Ï€ cm^2
c.)40Ï€ cm^2
5.)If a pyramid with a square base with its side measuring 6 m and its altitude of 3 m were circumscribed by a cone, what would be the ratio of the square base area to the circular base area?
a.)1 to π
b.)1 to 0.5Ï€
c.)1 to 2Ï€
6.)A magician has a conical hat that has a height of 1 m and a radius of 0.3 m. The hat does not fit his head. In order for the hat to fit perfectly, the radius needs to be 0.5 m. The magician can change the size of the hat with a spell, but the spell only changes the height of the hat, not the volume. After he performs the spell, what is the new height of the magician's hat if it has the desired radius of 0.5 m.
a.)0.03 m
b.)1.8 m
c.)0.36 m
1) For two similar shapes, the area ratio is the square of the length ratio for corresponding sides.
(1/2)^2 = __
2) Same principle as the last problem
3) I had to review the definition of prism. The ends have to be parallel, and the sides have to be parallelograms, not necessarily rectangles. The paralleogram-face requirement would make the edge lengths equal or "congruent", and parallel
4)2 pi R H = ___
5) The same as the ratio of the area of a square to that of a circumscribed circle. The height does not matter.
2/pi = 1/(0.5 pi)
6) R^2 H must remain constant
H2/H1 = (R1/R2)^2 = (3/5)^2 = 9/25
H2 = (9/25)*1 = 0.36
Thanks:)
5. b
1.) To find the ratio of the areas of two similar polygons, you need to know that the ratio of their corresponding side lengths squared is equal to the ratio of their areas. In this case, the length of one side of the smaller polygon is half the length of the larger polygon. So, the ratio of their corresponding side lengths squared is (1/2)^2 = 1/4. Therefore, the ratio of their areas is 1/4. The answer is b.) 1/4.
2.) If the length of the larger square is twice the length of the smaller square, then the area of the larger square is (2^2) = 4 times the area of the smaller square. Therefore, it would take 4 smaller squares to equal the area of 1 larger square. The answer is b.) 4.
3.) In any prism, the lateral edges are always congruent and parallel. The answer is b.) always true.
4.) The lateral area of a cylinder is given by the formula L = 2πrh, where r is the radius and h is the height. In this case, the radius is 5 cm and the height is 2 cm. Therefore, the lateral area is L = 2π(5 cm)(2 cm) = 20π cm^2. The answer is a.) 20π cm^2.
5.) The ratio of the areas of the square base to the circular base can be found by comparing their areas. The area of a square is given by the formula A = s^2, where s is the length of a side. In this case, the side of the square base is 6 m, so the area is A = (6 m)^2 = 36 m^2. The area of a circle is given by the formula A = πr^2, where r is the radius. In this case, the radius of the circular base is half the side length of the square base, so it is (6 m)/2 = 3 m. Therefore, the area of the circular base is A = π(3 m)^2 = 9π m^2. The ratio of the square base area to the circular base area is 36 m^2 : 9π m^2, which simplifies to 4 : π. The answer is b.) 1 to 0.5π.
6.) The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height. In this case, the radius of the hat is increasing from 0.3 m to 0.5 m, so the volume of the hat remains the same. Let's call the new height of the hat x. Then, the equation for the volume of the hat becomes (1/3)π(0.3 m)^2(1 m) = (1/3)π(0.5 m)^2(x m). This simplifies to (1/3)π(0.09)=(1/3)π(0.25)(x m). Canceling out the (1/3)π, we find that 0.09 = 0.25(x m). Solving for x, we get x = 0.36 m. The answer is c.) 0.36 m.