1)Two similar solids have a surface area of 48cm^2 and 108cm^2 respectively. Find the volume of the smaller solid if the bigger one has a volume of 162cm^3.
2)Otiende works for a coffee processing company as a sales man.He is paid on monthly basis as per agreement below;
a)A basic pay of sh 20000 per month.
b)A commission of 2% for goods sold upto a maximum of shs 200000
c)A commission of 4% for goods sold over sh200000 in that month he sold goods worth sh600000.Calculate his total pay for that month
3)In a figure, ABC is an equilateral triangle of side 15cm.Arcs AB,BC and AC are those of circles with centres C,A and B respectively.Calculate the area of the shaded region all round the edges.
1) Let's assume that the smaller solid has a volume of V cm^3.
Since the volume of a solid is directly proportional to the cube of its side length, we can set up a proportion using the surface areas:
(Volume of larger solid) : (Volume of smaller solid) = (Surface area of larger solid) : (Surface area of smaller solid)
162 : V = 108 : 48
Cross-multiplying and solving for V, we get:
162 * 48 = 108 * V
7776 = 108V
V = 7776 / 108
V = 72 cm^3
Therefore, the volume of the smaller solid is 72 cm^3.
2) Otiende's total pay for the month consists of his basic pay plus his commission. Let's calculate the commission first:
Commission on goods sold worth sh200,000 = 0.02 * 200,000 = sh4,000
Commission on goods sold over sh200,000 = 0.04 * (600,000 - 200,000) = sh16,000
Therefore, his total commission is sh4,000 + sh16,000 = sh20,000.
His total pay for the month is the sum of his basic pay and his commission:
Total pay = Basic pay + Commission
= 20,000 + 20,000
= sh40,000
Therefore, Otiende's total pay for that month is sh40,000.
3) The shaded region around the edges consists of three equal circular segments, each corresponding to an arc of one third of the circumference of their respective circles.
The radius of each circle can be found using the formula for the area of an equilateral triangle:
Area of equilateral triangle = (√3 / 4) * side^2
Substituting the given side length of 15 cm, we get:
Area of equilateral triangle = (√3 / 4) * 15^2 = 97.43 cm^2
Each radius will be one-third of the height of the equilateral triangle, which is also the radius of the circumcircle of the triangle. Using the formula for the radius of the circumcircle of an equilateral triangle:
Radius of circumcircle = (√3 / 3) * side
Substituting the given side length of 15 cm, we get:
Radius of circumcircle = (√3 / 3) * 15 = 8.66 cm
Since the angle of each circular segment is 120 degrees (360 degrees / 3), we can calculate the area of each segment using the formula:
Area of circular segment = (θ/360) * π * r^2
Substituting the given angle of 120 degrees and radius of 8.66 cm, we get:
Area of circular segment = (120/360) * π * (8.66)^2
= 1/3 * π * 75
= 25π
Since there are three equal circular segments, the total area of the shaded region is:
Total area of shaded region = 3 * 25π
= 75π
= 235.62 cm^2
Therefore, the area of the shaded region all around the edges is 235.62 cm^2.
1) Let's assume that the volume of the smaller solid is V. We know that the ratio of the surface areas of two similar solids is equal to the square of the ratio of their corresponding side lengths.
Since the surface area of the smaller solid is 48 cm², and the surface area of the larger solid is 108 cm², we can set up the following equation:
(48 / 108)² = V / 162
Simplifying the equation, we get:
(2 / 3)² = V / 162
4 / 9 = V / 162
Cross-multiplying, we have:
9V = 4 * 162
9V = 648
Dividing both sides by 9, we find that:
V = 72 cm³
Therefore, the volume of the smaller solid is 72 cm³.
2) Otiende's total pay for the month consists of his basic pay plus his commission. Let's calculate each component separately:
a) Otiende's basic pay is sh 20000.
b) Otiende's commission for goods sold up to sh 200000 is 2%. Since he sold goods worth sh 600000, we need to calculate the commission for the portion above sh 200000:
Commission = 2% * (600000 - 200000) = 2% * 400000 = sh 8000
However, since the commission is capped at sh 200000, we take the smaller value between sh 8000 and sh 200000:
Commission = sh 8000
c) Otiende's commission for goods sold above sh 200000 is 4%. Since he sold goods worth sh 600000, the commission for the portion above sh 200000 is:
Commission = 4% * (600000 - 200000) = 4% * 400000 = sh 16000
The total pay for the month is the sum of the basic pay and the commissions:
Total pay = Basic pay + Commission
Total pay = sh 20000 + sh 8000 + sh 16000
Total pay = sh 44000
Therefore, Otiende's total pay for that month is sh 44000.
3) To calculate the area of the shaded region all around the edges, we need to subtract the area of the equilateral triangle from the combined areas of the three arcs.
The area of an equilateral triangle with side length 15 cm is given by:
Area of equilateral triangle = (√3 / 4) * side length²
Area of equilateral triangle = (√3 / 4) * 15²
Area of equilateral triangle = (√3 / 4) * 225
Area of equilateral triangle = (15√3 / 4) * 15
Area of equilateral triangle = (225√3) / 4
Now let's calculate the combined areas of the three arcs. The central angle of each arc is 120° since it's a third of a full circle. The radius of each arc is the same as the side length of the equilateral triangle, which is 15 cm.
Area of a sector = (central angle / 360°) * π * radius²
Area of each sector = (120 / 360) * π * 15²
Area of each sector = (1 / 3) * π * 225
Area of each sector = (π / 3) * 225
Since there are three of these sectors, the combined area is:
Combined area of three sectors = 3 * (π / 3) * 225
Combined area of three sectors = π * 225
Now, we can calculate the area of the shaded region by subtracting the area of the equilateral triangle from the combined area of the three sectors:
Area of shaded region = Combined area of three sectors - Area of equilateral triangle
Area of shaded region = π * 225 - (225√3) / 4
Therefore, the area of the shaded region all around the edges is π * 225 - (225√3) / 4.