Test: Geometry - GMAT MCQ

A solid brick is made up of small block of side 1.
small block of side 1.
The large brick is made up of 12 small cubes.
So, length of the large brick = 3
width of the large brick = 2
height of the large brick = 2
∴ total surface area of large brick
=  total surface area of cuboid
= 2(2(3 × 2 + 2 × 2 + 3 × 2) 
= 2(6 + 4 + 6)
= 2(16)
= 32
⇒ surface area of large brick 32
Total surface area all 12 smaller
bricks = 10(6 × 1)
= 72
∴ Total surface are of all 12
smaller bricks − Total surface area of large brick
⇒ 72 − 32 = 40
⇒ Total surface area of all the small bricks is 40 greater than the surface area of a large brick.
option (B) 40 is correct answer.

Given that the front wheels have a circumference of 1 meter, we can deduce that their radius is half of this circumference, which is 1/2 meter or 0.5 meters.

Let's denote the radius of the rear wheels as R. According to the problem, the radius of the front wheels is half the radius of the rear wheels. Therefore, we can express the radius of the front wheels in terms of R as follows:

Front wheel radius = 0.5R

We know that the circumference of a circle is given by the formula C = 2πr, where C represents the circumference and r represents the radius. Using this formula, we can express the circumference of the front wheels as:

1 meter = 2π(0.5R) 1 = πR R = 1/π

Now we have the radius of the rear wheels, which is 1/π meters.

To calculate the number of revolutions made by each rear wheel, we can use the formula:

Number of revolutions = Distance traveled / Circumference of the wheel

The distance traveled is given as 1 kilometer, which is equal to 1000 meters. The circumference of a wheel is given by the formula C = 2πr.

Number of revolutions made by each rear wheel = 1000 meters / (2π(1/π)) = 1000 meters / (2π/π) = 1000 meters / 2 = 500 revolutions

Therefore, each rear wheel would make 500 revolutions.

The formula for the circumference of a circle is C = 2πr, where r represents the radius of the circle.

For the second circle: C = 2πr

For the first circle: 2C = 2(2πr) = 4πr

Given that the total circumference of the two circles is 36, we can write the equation: C + 2C = 36 3C = 36 C = 36/3 C = 12

Substituting C = 12 back into the equation for the second circle: 2πr = 12 r = 12/(2π) r ≈ 1.91

For the first circle, the radius is double that of the second circle: 2r ≈ 2(1.91) ≈ 3.82

The approximate sum of their two radii is: 1.91 + 3.82 ≈ 5.73

Therefore, the correct answer is approximately 5.7, which corresponds to option (A).

Let's assume the distance between Cedar Rapids and the point where Mr. Dinkle overtakes Mr. Smitherly is denoted as "D" miles.

Since Mr. Smitherly leaves Cedar Rapids at 8 a.m. and drives at an average speed of 50 miles per hour, he will have traveled for 0.5 hours (30 minutes) when Mr. Dinkle starts driving.

During this time, Mr. Smitherly will have covered a distance of: Distance = Speed * Time Distance = 50 mph * 0.5 hours Distance = 25 miles

Therefore, when Mr. Dinkle starts driving at 8:30 a.m., Mr. Smitherly is 25 miles ahead.

Now, let's calculate how long it will take for Mr. Dinkle to catch up to Mr. Smitherly. We can set up the following equation:

Time = Distance / Speed

Since both Mr. Smitherly and Mr. Dinkle are driving at constant speeds, we can write:

Time taken by Mr. Smitherly = Time taken by Mr. Dinkle

(D - 25 miles) / 50 mph = D / 60 mph

Now, we can solve this equation to find the value of D.

Cross-multiplying the equation gives:

60(D - 25) = 50D

60D - 1500 = 50D

10D = 1500

D = 1500 / 10

D = 150 miles

So, the point where Mr. Dinkle overtakes Mr. Smitherly is 150 miles from Cedar Rapids.

To find out when Mr. Dinkle will overtake Mr. Smitherly, we need to calculate the time it takes for Mr. Dinkle to travel 150 miles.

Time = Distance / Speed Time = 150 miles / 60 mph Time = 2.5 hours

Therefore, Mr. Dinkle will overtake Mr. Smitherly 2.5 hours after he starts driving at 8:30 a.m.

Adding this time to the starting time gives us:

8:30 a.m. + 2.5 hours = 11:00 a.m.

Therefore, the correct answer is (C) Mr. Dinkle will overtake Mr. Smitherly at 11:00 a.m.

Suppose there are c columns and there are r rows
Original Situation
So, Number of tiles = c * r = 70
Also. Reach column has r tiles and each row has c tiles

New Situation
Number of tiles in each column is r-2 and number of tiles in each row is c + 4
So, number of rows = r-2 and number of columns is c + 4
So, Number of tiles = (r - 2)*(c + 4) = 70

Comparing both of them we get
c*r = (r - 2)*(c + 4)
=> 4r -2c = 8
c = 2r - 4
Putting it in c * r = 70
(2r - 4) * r = 70
2r² - 4r - 70=0
r² - 2r - 35 = 0
r² -7r +5r - 35 = 0
r = -5,7
r cannot be negative so r = 7
and c = 10
So, Answer will be C

Let's assume the original length of the rectangular garden is L feet and the original width is W feet.
Therefore, the original area of the garden is A = L * W.

According to the problem, if we increase the length by 7.5 feet, the new length will be L + 7.5 feet.
Similarly, if we increase the width by 5 feet, the new width will be W + 5 feet.

In the first scenario, the area of the garden after increasing the length is (L + 7.5) * W, which is A + 7.5W.

In the second scenario, the area of the garden after increasing the width is L * (W + 5), which is A + 5L.

We are given that either of these scenarios results in an increase of 150 square feet in the area of the garden.
Therefore, we can set up the equation:

A + 7.5W = A + 150 (from the first scenario) A + 5L = A + 150 (from the second scenario)

Simplifying these equations, we get: 7.5W = 150 5L = 150

Dividing both sides of the first equation by 7.5, we get: W = 20

Dividing both sides of the second equation by 5, we get: L = 30

Now, we can substitute the values of L = 30 and W = 20 into the original area equation A = L * W: A = 30 * 20 A = 600

Therefore, the area of the garden is 600 square feet. The correct answer choice is (A) 600.

The given problem asks about the number of diagonals in a regular 11-sided polygon.
A diagonal is a line segment that connects two non-adjacent vertices of a polygon.
To find the number of diagonals, we can use the formula:

Number of diagonals = n * (n - 3) / 2

Where 'n' is the number of sides of the polygon. In this case, n = 11.
Plugging the value into the formula, we have:

Number of diagonals = 11 * (11 - 3) / 2 = 11 * 8 / 2 = 88 / 2 = 44

Therefore, a regular 11-sided polygon contains 44 diagonals. The correct answer is (d) 44.

The formula for the number of diagonals in a polygon is n(n-3)/2 where n is the number of sides of the polygon.

If each vertex is the intersection of exactly three diagonals, then each vertex must correspond to a polygon with at least four sides.

When n = 4, there is only one diagonal, so no vertex can be the intersection of three diagonals.

When n = 5, there are five diagonals, so each vertex can be the intersection of at most two diagonals.

When n = 6, there are nine diagonals, so it is possible for each vertex to be the intersection of three diagonals.

Therefore, the answer is (D) 6.

We have a rectangle with integer side lengths and a perimeter of 10. Let's find the possible dimensions of this rectangle:

The perimeter of a rectangle is given by the formula P = 2l + 2w, where l is the length and w is the width.
In this case, P = 10, so we have the equation 10 = 2l + 2w.
The possible dimensions of this rectangle are (1, 4) and (2, 3), as these are the only pairs of integers that satisfy the equation.
Now, let's analyze the number of rectangles we can cut from the given piece of paper with dimensions 24 and 60:

1. When the dimensions of the rectangles are (1, 4):
We can cut 60 rectangles along the width (24/1 = 24) and 15 rectangles along the length (60/4 = 15).
This gives us a total of 60 x 15 = 900 rectangles.

2. When the dimensions of the rectangles are (2, 3):
We can cut 30 rectangles along the width (24/2 = 12) and 20 rectangles along the length (60/3 = 20).
This gives us a total of 30 x 20 = 600 rectangles.

Comparing the two cases, we can cut more rectangles with the dimensions (1, 4). However, we need to check if the total number of rectangles (900) is divisible by the number of rectangles we can cut along the width (24).

900/24 = 37.5, which is not an integer.
To find the largest integer divisible by 24, we can multiply 24 by the largest integer less than 37.5, which is 37.
So, the largest number of rectangles we can cut is 24 x 37 = 888.
However, this is not among the given options. We need to check if any of the given options can be formed by a valid arrangement of rectangles.
Option D (360) can be achieved by cutting 15 rectangles along the width and 24 rectangles along the length (15 x 24 = 360).

Therefore, the greatest number of rectangles that can be cut from the given piece of paper is Option D.

Heron's formula states that the area (A) of a triangle with side lengths a, b, and c is given by:

A = √(s(s - a)(s - b)(s - c))

where s is the semiperimeter of the triangle, defined as:

s = (a + b + c) / 2

In this case, the side lengths are:
ST = 17
TV = 17
SV = 16

Let's calculate the semiperimeter first:

s = (ST + TV + SV) / 2
= (17 + 17 + 16) / 2
= 50 / 2
= 25

Now we can substitute the values into Heron's formula to find the area:

A = √(s(s - ST)(s - TV)(s - SV))
= √(25(25 - 17)(25 - 17)(25 - 16))
= √(25(8)(8)(9))
= √(25 * 8² * 9)
= √(25 * 64 * 9)
= √(14400)
= 120

Therefore, the area of triangle STV is 120 square units.