1. x/5
_______ _____
y/20|_______| ________________| | Y
x/25 y/4 |_____________________|
Tile X
Bathroom
Using the diagram above(that i tried to replicate out of ascii), if rectangular tiles are used to completely cover the L shaped bathroom floor without overlap how many rectangular tiles are needed?
F)125
G)150
H)175
J)200
K)225
2. If x(b-c)=y+x and 2b=3c=7, then y/x=
A)1/7
B)1/6
C)1/3
D)5/7
E)2/3
3. Three small circles of equal area are on line segment AB with centers C,D, and E. Each of the three small circles is tangent to two other circles. The area of the large circle with center D is 900 pi. What is the circumference of the circle with center C?
Note: I cant replicate the diagram cuz its a cricle. But the diagram is a large circle with a line across the middle and three circles(labeled C,D and E in that order) going straight through the line with dots in the middle.
A)5 pi
B)10 pi
C)20 pi
D)40 pi
E) 60 pi
I know the diagram is a bit mucked up. So im gonna say the lengths of the shaped. The tile is a rectangle with a length of x/25 and the width is y/20. The bathroom is harder to describe but i'll try. Its an L and the top of the L 's length is y/4 and the bottom is just y. The left side of the L is x and the bottom right side is x/5.
1. To determine how many rectangular tiles are needed to cover the L-shaped bathroom floor without overlap, we need to find the total area of the bathroom floor and then divide it by the area of one rectangular tile.
Looking at the diagram, the bathroom floor can be divided into two rectangular sections: one with dimensions x/5 by y/20, and the other with dimensions x/25 by y/4.
To find the area of the first rectangle, we multiply the length by the width: (x/5) * (y/20) = xy/(5*20).
To find the area of the second rectangle, we multiply the length by the width: (x/25) * (y/4) = xy/(25*4).
Now, we add the areas of the two rectangles to find the total area of the bathroom floor: xy/(5*20) + xy/(25*4).
To simplify the expression, we find the least common multiple of 5, 20, and 25, which is 100. Then we can rewrite the expression as: xy/100 + xy/100.
Combining the two terms with common denominators, we have: (xy + xy)/100 = (2xy)/100 = xy/50.
Now, to find the number of rectangular tiles needed, we divide the total area of the bathroom floor by the area of one rectangular tile. Let's assume the rectangular tile has dimensions a by b.
The area of one rectangular tile is simply a * b.
Dividing the total area of the bathroom floor (xy/50) by the area of one rectangular tile (a * b), we get: (xy/50) / (a * b) = xy / (50 * a * b).
Since the dimensions of the rectangular tiles are not provided in the question, we cannot calculate the exact number of tiles needed. Therefore, we cannot determine the correct option out of the given choices (F, G, H, J, K).
2. Given the equation x(b-c) = y + x and the values 2b = 3c = 7, we need to find the value of y/x.
First, we can simplify the equation x(b-c) = y + x by expanding the expression on the left side: xb - xc = y + x.
Since we're given 2b = 3c = 7, we can substitute these values into the equation: x(2) - x(7/3) = y + x.
Simplifying further, we have: 2x - (7/3)x = y + x.
Combining like terms, we get: (2 - 7/3)x = y + x.
To simplify the left side of the equation, we find a common denominator for 2 and 7/3, which is 3. This gives us: (6/3 - 7/3)x = y + x.
Now, we have: (-1/3)x = y + x.
To isolate y/x, we divide both sides of the equation by x: (-1/3)x / x = (y + x) / x.
Simplifying further, we get: -1/3 = y/x + 1.
To isolate y/x, we subtract 1 from both sides of the equation: (-1/3) - 1 = y/x + 1 - 1.
Simplifying, we have: -4/3 = y/x.
Therefore, y/x is equal to -4/3.
Neither of the given answer choices (A, B, C, D, E) matches -4/3.
3. In the given diagram, three small circles of equal area are on line segment AB with centers C, D, and E. The area of the large circle with center D is 900π. We need to find the circumference of the circle with center C.
Since the three small circles are tangent to two other circles each, they are externally tangent.
Let's assume the radius of each small circle is r. Since the small circles are equal in area, we can infer that the radius of the large circle is 2r. This is because the area of a circle is proportional to the square of its radius.
Given that the area of the large circle (with center D) is 900π, we can find the radius by using the formula for the area of a circle: Area = πr^2.
Substituting the given values, we have: 900π = π(2r)^2.
Simplifying the equation, we get: 900 = 4r^2.
Dividing both sides of the equation by 4, we have: 225 = r^2.
Taking the square root of both sides, we find: r = 15.
Now that we know the radius of the small circles, we can calculate the diameter of the small circle with center C by adding the diameter of one small circle (2r) to the diameter of the large circle (2 * 2r).
The diameter of the small circle with center C is: 2r + 2 * 2r = 2r(1 + 2) = 6r.
Substituting the value of r (15), we have: 6(15) = 90.
Therefore, the circumference of the circle with center C is 90π.
None of the given answer choices (A, B, C, D, E) matches 90π, so the correct answer choice is not provided.