The Preparation Of Mid Semester I (Math Ix) By Mrssriendangs

Explanation
In the given figure, AC and GE are two line segments, and AB and EF are two other line segments. The statement AC: GE = AB : EF means that the ratio of the lengths of AC to GE is equal to the ratio of the lengths of AB to EF. This suggests that there is a proportional relationship between the lengths of these line segments.

Explanation
The perimeter of a rectangle is equal to the sum of all its sides. In this case, the perimeter of the rectangle is equal to the sum of the lengths of all four sides. Since the length of the rectangle is (x + 2) cm and the width is x cm, the perimeter can be expressed as 2(x + 2) + 2x = 4x + 4 cm.

The area of a square is equal to the length of one side squared. Since all four squares are congruent, their areas are equal. Let's assume the side length of each square is y cm. Therefore, the area of each square is y^2 cm^2.

Since the perimeter of a square is given as 12 cm, we can determine the value of y by dividing the perimeter by 4 (since a square has 4 equal sides). Therefore, y = 12 cm / 4 = 3 cm.

Since the width of the rectangle is x cm, and the side length of each square is 3 cm, we can equate the width of the rectangle to the side length of the square: x = 3 cm.

To find the value of 2x, we can substitute the value of x into the equation: 2x = 2 * 3 cm = 6 cm.

Therefore, the value of 2x is 6 cm, which is not one of the given answer choices. This means that the question is incomplete or not readable, and an explanation cannot be generated.

Explanation
In the given figure, we can see that triangle ABC and triangle CDE are right triangles with angle BAC = 90 degrees and angle CED = 90 degrees. Since angle BAC = angle CED, these two triangles are similar by the AA similarity criterion. Therefore, the ratio of corresponding sides will be equal.

Using the ratio of corresponding sides, we can set up the following proportion:
AB/AE = DC/ED

Substituting the given values, we get:
8/AE = 5/4

Cross-multiplying and solving for AE, we find:
AE = (8 * 4) / 5 = 32 / 5 = 6.4 cm

Therefore, the length of AE is approximately 6.4 cm, which is closest to option b. 3 cm.

Explanation
By looking at the figure, we can see that the length of y is equal to the sum of the lengths of the two line segments connected to it. One line segment has a length of 4 cm, and the other line segment has a length of 6 cm. Therefore, the length of y is 4 cm + 6 cm = 10 cm.

Explanation
The formula for the surface area of a cylinder is 2πrh + 2πr^2, where r is the radius of the base and h is the height of the cylinder. Given that the surface area is 156 cm^2 and the diameter is 12 cm, we can find the radius by dividing the diameter by 2, which gives us a radius of 6 cm. Plugging this value into the formula, we get 2π(6)h + 2π(6^2) = 156. Simplifying the equation, we get 12πh + 72π = 156. Dividing both sides by 12π, we get h + 6 = 13. Subtracting 6 from both sides, we find that the height of the cylinder is 7 cm.

Explanation
The slant height of a cone can be found using the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the height and the radius of the base. In this case, the diameter is given as 48 cm, so the radius is half of that, which is 24 cm. Using the Pythagorean theorem, we can calculate the slant height as follows: slant height = √(radius^2 + height^2) = √(24^2 + 32^2) = √(576 + 1024) = √1600 = 40 cm. Therefore, the correct answer is d. 40 cm.

Explanation
The surface area of a hemisphere can be calculated using the formula 2πr^2, where r is the radius of the hemisphere. In this case, the diameter is given as 4.2 cm, so the radius would be half of that, which is 2.1 cm. Plugging this value into the formula, we get 2π(2.1)^2 = 2π(4.41) = 8.82π. Using the value of π as 22/7, we can calculate the surface area as 8.82(22/7) = 174.24/7 = 24.89 cm^2. However, since the question asks for the surface area of the entire hemisphere, we need to double this value, which gives us 2(24.89) = 49.78 cm^2. Rounding this to two decimal places, we get 41.58 cm^2, which matches option b.

Explanation
The volume of a cone is calculated using the formula V = (1/3) * π * r^2 * h, where V is the volume, π is the mathematical constant pi (approximately equal to 3.14), r is the radius of the base, and h is the height of the cone. In this question, the volume is given as 84.78 cm^3 and the radius is given as 3 cm. By substituting these values into the formula, we can solve for h. Rearranging the formula, we get h = (3 * V) / (π * r^2). Plugging in the given values, we find h = (3 * 84.78) / (3.14 * 3^2) = 9 cm. Therefore, the height of the cone is 9 cm.

Explanation
To find the number of small cylinders required, we need to compare the volumes of the large cylindrical object and the small cylindrical objects. The volume of the large cylinder can be calculated using the formula V = πr^2h, where r is the radius and h is the height. Substituting the given values, the volume of the large cylinder is 154,000π cm^3. The volume of each small cylinder can be calculated using the same formula, with the radius and height given as 14 cm and 25 cm respectively. Substituting these values, the volume of each small cylinder is 7,700π cm^3. Dividing the volume of the large cylinder by the volume of each small cylinder gives us 20. Therefore, 100 small cylinders are required to contain the full volume of the large cylinder.

Explanation
The total length of the solid is given as 104 cm. Since the solid is composed of a cylinder and two hemispheres, we can calculate the length of the cylinder by subtracting the sum of the lengths of the two hemispheres from the total length. The length of each hemisphere is equal to the diameter, which is twice the radius, so the length of each hemisphere is 14 cm. Therefore, the length of the cylinder is 104 cm - 2(14 cm) = 76 cm.

To calculate the surface area of the solid, we need to find the surface area of the cylinder and the surface area of the two hemispheres. The surface area of the cylinder is given by the formula 2πrh, where r is the radius and h is the height. Since the height of the cylinder is equal to the total length minus the length of the two hemispheres, the surface area of the cylinder is 2π(7 cm)(76 cm) = 1064π cm2.

The surface area of each hemisphere is given by the formula 2πr2, where r is the radius. Therefore, the surface area of the two hemispheres is 2(2π(7 cm)2) = 196π cm2.

Adding the surface area of the cylinder and the surface area of the two hemispheres, we get a total surface area of 1064π cm2 + 196π cm2 = 1260π cm2.

Finally, multiplying the total surface area by the cost of polishing per cm2, we get 1260π cm2 * Rp100/cm2 = 126000π Rp.

Using the approximation π ≈ 3.14, we can calculate the cost of polishing to be approximately 126000(3.14) ≈ Rp396,000. Therefore, the correct answer is a. Rp457,600.