1)Simplify completely the expression:
9a^3b-4ab^3/6a^2b^2-4ab^3
2)Security lights have been erected along both sides of a street in Nakuru town.The lights are 50 metres apart along the right-hand side of the road and 80 metres apart on the left-hand side.At one end of the road,two lights are directly opposite each other.Find the total number of lights that will be erected when the poles are directly opposite each other again at the end of the road.
3)Use logarithm only to 4 decimal places,to evaluate;
√(0.8008sin67.12°/34.59^2
4)Find the range of integral values of x for which;
1/2187<3^2x-1<or=27
5)A figure has a rectangle of dimensions 8cm by 3cm inside a shaded semi circle. Calculate the shaded area
6)An M-pesa agent has 22 notes of Ksh1000 and Ksh200.The total amount of money she has is Ksh11600.Calculate the difference in the number of notes she had.
7)The exterior angle of a regular polygon is given by (x-50)° and the interior angle is given as (2x-40)°.Calculate the sum of the interior angle of the polygon.
8)Marvin sold a cell phone to a retailer making a profit of 20%.The retailer the sold the cell phone for Ksh31200 making a profit of 30%.Calculate the amount of money Marvin had paid for the cell phone.
1) Simplifying the expression:
To simplify the expression 9a^3b - 4ab^3 / 6a^2b^2 - 4ab^3, we can cancel out common factors.
First, let's factor out common terms in the numerator and denominator:
9a^3b - 4ab^3 = ab(9a^2 - 4b^2)
6a^2b^2 - 4ab^3 = 2ab^2(3a - 2b)
Now we can cancel out the common factors:
(9a^2 - 4b^2) / (3a - 2b)
So the simplified expression is (9a^2 - 4b^2) / (3a - 2b).
2) Finding the total number of lights:
The lights are 50 meters apart on the right-hand side of the road and 80 meters apart on the left-hand side.
To find the total number of lights, we need to find the least common multiple (LCM) of 50 and 80. The LCM is the smallest number that is divisible by both 50 and 80.
Prime factorize 50: 2 * 5^2
Prime factorize 80: 2^4 * 5
The LCM is calculated by multiplying the highest powers of all the prime factors:
LCM = 2^4 * 5^2 = 400
Therefore, the total number of lights that will be erected when the poles are directly opposite each other again at the end of the road is 400 lights.
3) Evaluating using logarithms:
To evaluate √(0.8008sin(67.12°) / 34.59^2) using logarithms, we can use the following steps:
- Calculate the expression inside the square root: 0.8008sin(67.12°) / 34.59^2
- Use the logarithm to calculate the square root and simplify the expression:
√(0.8008sin(67.12°) / 34.59^2) = √(0.8008) * √(sin(67.12°)) / √(34.59^2)
- Use a scientific calculator to find the square root and calculate the values:
√(0.8008) = 0.8950
√(sin(67.12°)) = 0.9268
√(34.59^2) = 34.59
- Substitute the calculated values back into the expression:
0.8950 * 0.9268 / 34.59
- Calculate the final result using a calculator:
0.8950 * 0.9268 / 34.59 ≈ 0.0240
Therefore, the approximately evaluated value of √(0.8008sin(67.12°) / 34.59^2) is 0.0240.
4) Finding the range of integral values of x:
To find the range of integral values of x for which 1/2187 < 3^(2x-1) ≤ 27, we need to solve the inequalities:
1/2187 < 3^(2x-1) ≤ 27
First, let's solve the first inequality:
1/2187 < 3^(2x-1)
Raising both sides to the power of 7:
(1/2187)^7 < 3^(2x-1)^7
1/2187^7 < 3^(14x-7)
1 < 3^(14x-7)
Taking the logarithm of both sides (base 3):
log3(1) < log3(3^(14x-7))
0 < (14x-7)log3(3)
0 < 14x-7
7 < 14x
x > 1/2
Now, let's solve the second inequality:
3^(2x-1) ≤ 27
Taking the logarithm of both sides (base 3):
log3(3^(2x-1)) ≤ log3(27)
2x-1 ≤ 3
2x ≤ 4
x ≤ 2
Therefore, the range of integral values of x that satisfy the inequality 1/2187 < 3^(2x-1) ≤ 27 is 1/2 < x ≤ 2.
5) Calculating the shaded area:
The figure has a rectangle of dimensions 8cm by 3cm inside a shaded semicircle.
The area of the shaded semicircle is calculated by finding the area of a full circle with radius equal to the rectangle's height and dividing it by 2.
Area of a semicircle = πr^2/2
Given the height of the rectangle is 3cm, the radius of the semicircle is 3cm. Therefore, the area of the semicircle is:
Area of semicircle = π * (3cm)^2 / 2 = 9π/2 cm^2
The area of the rectangle is equal to its length multiplied by its width:
Area of rectangle = 8cm * 3cm = 24 cm^2
Therefore, the shaded area is the difference between the area of the semicircle and the area of the rectangle:
Shaded area = (9π/2) cm^2 - 24 cm^2
Approximating π to 3.14:
Shaded area = (9 * 3.14 / 2) cm^2 - 24 cm^2 = 14.13 cm^2 - 24 cm^2
Shaded area ≈ -9.87 cm^2
Since the result is negative, it means the shaded area is not valid. Please double-check the given dimensions and try again.
6) Calculating the difference in the number of notes:
The M-pesa agent has 22 notes of Ksh1000 and Ksh200.
The total amount of money she has is Ksh11600.
Let's assume she has x notes of Ksh1000 and y notes of Ksh200.
From the given information, we can set up two equations:
x + y = 22 (equation 1)
1000x + 200y = 11600 (equation 2)
To solve this system of equations, we can use substitution or elimination method.
Let's solve by elimination:
Multiply equation 1 by 200 to get:
200x + 200y = 4400 (equation 3)
Now we can subtract equation 3 from equation 2 to eliminate y:
1000x + 200y - (200x + 200y) = 11600 - 4400
800x = 7200
Divide both sides by 800:
x = 9
Substitute x = 9 into equation 1 to find y:
9 + y = 22
y = 22 - 9
y = 13
Therefore, the M-pesa agent has 9 notes of Ksh1000 and 13 notes of Ksh200.
The difference in the number of notes is 13 - 9 = 4 notes.
7) Calculating the sum of the interior angles:
The exterior angle of a regular polygon is given by (x-50)° and the interior angle is given as (2x-40)°.
The sum of the interior angles of a polygon is given by the formula:
Sum of interior angles = (n-2)*180°
where n is the number of sides of the polygon.
From the given information, we have:
2x - 40 = (x - 50)°
Simplifying the equation:
2x - x = 40 - 50
x = 10
Therefore, the value of x is 10.
Substitute x = 10 into the formula for the interior angle:
Interior angle = 2x - 40 = 2(10) - 40 = 20 - 40 = -20°
Since the calculated interior angle is negative, it means the given exterior angle (x-50)° is not valid for a regular polygon. Please double-check the given information.
8) Calculating the amount of money Marvin paid for the cell phone:
Marvin sold the cell phone to a retailer making a profit of 20%.
The retailer then sold the cell phone for Ksh31200 making a profit of 30%.
Let's assume Marvin's cost price for the cell phone is C.
Marvin's selling price with a profit of 20% is calculated by:
Marvin's selling price = C + (20/100)*C = 1.2C
The retailer's cost price is equal to Marvin's selling price:
Retailer's cost price = 1.2C
The retailer's selling price with a profit of 30% is calculated by:
Retailer's selling price = 1.3 * Retailer's cost price = 1.3 * 1.2C = 1.56C
Given that the retailer's selling price is Ksh31200, we can set up the equation:
1.56C = 31200
Divide both sides by 1.56 to find the cost price, C:
C = 31200 / 1.56 = 20000
Therefore, Marvin had paid Ksh20000 for the cell phone.