1. Juanita has 180 feet of fencing that she intends to use to build a rectangular play area for her dog. She wants the play area to enclose at least 1800 square feet. What are the possible widths of the play area?
2. A bicycle maker sold 300 bicycles last year at a profit of $300 each. The maker wants to increase the profit margin this year, but predicts that each $20 increase in profit will reduce the number of bicycles sold by 10. How many $20 increases in profit can the maker add in and expect to make a total profit of at least $100,000?
3. The current in a circuit is amps (I), and the impedance is ohms (Z). What is the voltage? (Hint: use the formula
4. Write two complex numbers whose product is 10.
I will do #5 for you
please show some ideas or work that you tried for the others
let the two numbers be a+bi and a-bi
then (a+bi)(a-bi)= 10
a^2 - b^2i^2 = 10
a^2 + b^2 = 10 .... because i^2 = -1
let a=1, then b^2 = 9
b = 3 or b= -3
so let's pick a=1 and b-3
two complex numbers whose product are 10 are
1 + 3i and 1-3i
1. To find the possible widths of the play area, we need to consider that the sum of all the sides of a rectangle is equal to 2 times the length plus 2 times the width. We also need to find the length and width that will enclose at least 1800 square feet.
Let's start by assigning variables:
Let L represent the length of the play area.
Let W represent the width of the play area.
We know that the sum of all the sides of the rectangle is equal to 180 feet, so we can write an equation:
2L + 2W = 180
Next, we know that the area of a rectangle is equal to the length multiplied by the width:
L * W ≥ 1800
Now we have a system of equations. We can solve this system to find the possible values of the width.
2. To find out how many $20 increases in profit the maker can add and still make a total profit of at least $100,000, we need to consider the relationship between the number of bicycles sold and the profit per bicycle.
Let's assign variables:
Let x represent the number of $20 increases in profit.
Let p be the profit per bicycle.
We know that each $20 increase in profit reduces the number of bicycles sold by 10. So, the equation relating the number of bicycles sold (B) to the number of $20 increases in profit is:
B = 300 - 10x
Now we need to find the total profit by multiplying the number of bicycles sold by the profit per bicycle:
Total profit = B * p
We want the total profit to be at least $100,000, so we can write the equation:
300p - 10px ≥ 100,000
Now we have an equation with one unknown variable (x). We can solve this equation to find the number of $20 increases in profit the maker can add.
3. To find the voltage (V) in a circuit given the current (I) and impedance (Z), we can use Ohm's Law, which states that V = I * Z. So, we simply multiply the current by the impedance to get the voltage.
4. To write two complex numbers whose product is 10, we can use the fact that the product of two complex numbers is equal to the product of their magnitudes, multiplied by the sum of their angles.
Let's assume the numbers are z1 and z2. We can write them in polar form as follows:
z1 = r1 * (cosθ1 + i sinθ1)
z2 = r2 * (cosθ2 + i sinθ2)
where r1 and r2 are the magnitudes of z1 and z2, and θ1 and θ2 are their angles.
We want the product of z1 and z2 to be 10. So, we have:
z1 * z2 = 10
(r1 * r2) * (cos(θ1 + θ2) + i sin(θ1 + θ2)) = 10
Now, we have an equation with two unknown variables, r1 * r2 and (θ1 + θ2). We can choose any appropriate values for these variables to write two complex numbers whose product is 10.