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.
4. what about 3+i, 3-i ? or -sqrt10 i, +sqrt10 i
or 1 + 3i,1-3i
on 1. You know the perimeter (180=2L+2w) and the area 1800<L*W
Solve by substution.
1. To find the possible widths of the play area, we can set up an equation based on the given information. Let's assume the width of the rectangular play area is "w" feet.
The perimeter of a rectangle is given by the formula P = 2(l + w), where "l" is the length. In this case, we only have the total fencing length, which is 180 feet. So we can write the equation as:
2(l + w) = 180
We also know that the area of a rectangle is given by the formula A = l * w. In this case, we want the play area to enclose at least 1800 square feet. So we can write the equation as:
l * w >= 1800
To find the possible widths, we can solve these two equations simultaneously.
2(l + w) = 180 --> l + w = 90 --> l = 90 - w
Substituting this value of "l" in the area equation:
(90 - w) * w >= 1800
Now, we can find the possible widths by solving this inequality.
2. To determine how many $20 increases in profit can the maker add in and expect to make a total profit of at least $100,000, we need to set up an equation based on the given information.
Let's assume the number of $20 increases in profit is "n". Each increase will reduce the number of bicycles sold by 10, so the number of bicycles sold can be calculated as 300 - 10n.
The new profit for each bicycle will be $300 + $20n.
To find the total profit, we multiply the number of bicycles sold by the new profit for each bicycle:
Total profit = (300 - 10n) * ($300 + $20n)
We want the total profit to be at least $100,000, so we set up the following inequality:
(300 - 10n) * (300 + 20n) >= 100,000
Solve this inequality to find the maximum value of "n" that satisfies the condition.
3. The voltage in a circuit can be found using Ohm's Law, which states that voltage (V) is equal to the current (I) multiplied by the impedance (Z), or V = I * Z.
To find the voltage, you need to know the values of current and impedance. Once you have those values, you can calculate the voltage using the formula V = I * Z.
4. To write two complex numbers whose product is 10, we can set up an equation and solve for the complex numbers.
Let's assume the two complex numbers are a + bi and c + di, where "a", "b", "c", and "d" are real numbers.
Their product is given by (a + bi)(c + di). To find the complex numbers whose product is 10, we can set up the equation:
(a + bi)(c + di) = 10
Expand the equation using the distributive property and equate the real and imaginary parts of the equation:
ac + adi + bci + bdi^2 = 10
Combine like terms and note that i^2 is equal to -1:
(ac - bd) + (ad + bc)i = 10
From this equation, we can match the real and imaginary parts with the corresponding coefficients.
(ac - bd) = 10 (equation 1)
(ad + bc) = 0 (equation 2)
Solve these two equations simultaneously to find the values of "a", "b", "c", and "d".