College Algebra Final Practice Test KEY - ALL Questions Name____________________________________

1.       Solve (2x + 5)(x - 2) = x(2x-2). Show all work!

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x = 10/3 (x-squared terms cancel)
1.       Solve the absolute value equation shown below. Show all work!

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x=2 or x=-5
1.       The price of an item is given by the equation P = 0.018X + 300 where X= number of items sold and P = the price. If the price is $400, how many items are sold, rounded to the nearest whole number? Show all work!

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x=5556
1.       Solve the equation given below. Show all work!

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x=-6
2.       Solve 4x2 - 10x = 24 Show all work!

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x=4 or x=-3/2
2.       Solve (3x + 1)2 = 7 Show all work!

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2.       Solve 2x2 = 5x + 3 Show all work!

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x=3 or x = -1/2
2.       Solve (2x + 1)2 - 1 = 0 Show all work!

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x=0 or x=-1
3.       Solve 2x2 + 3x - 4 = 0 by completing the square. You MUST complete the square and show all work!

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3.       Solve 3x2 + 2x - 6 = 0 by completing the square. You MUST complete the square and show all work!

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3.       Solve 2x2 + 3x - 8 = 0 by completing the square. You MUST complete the square and show all work!

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3.       Solve 5x2 + 4x - 10 = 0 by completing the square. You MUST complete the square and show all work!

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4.       Solve 2x3 - 32x = 0. Show All Work!

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x=0 or x=4 or x=-4
4.       Solve the equation given below. Show all work!

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4.       Solve 3x4 - 6x3 + 3x2 = 0. Show All Work!

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x=0 or x=1
4.       Solve the equation given below. Show all work!

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5.       Solve the equation given below. Show the Algebraic Solution!

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x=7 (toss out the answer x=2)
5.       Solve the equation given below. Show the Algebraic Solution!

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x=6 (toss out the answerx=3)
5.       Solve 2x4 + 4x2 - 1 = 0 Hint: Let u=x2 and solve for u first. Show All Work!

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5.       Solve 3x4 - 8x2 + 1 = 0 Hint: Let u=x2 and solve for u first. Show All Work!

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6.       The length L of a skid mark left by a car when the brakes are suddenly applies varies directly as the square of the speed S of the car. If a car traveling S=30 MPH leaves a L=50 ft skid mark, how fast would this car have to go in order to leave a 200 ft skidmark? Show ALL Work!

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Using model L=kS2, k=0.056 and S=59.8 MPH when L=200 ft
6.       An object that is dropped travels a distance D that varies directly with the square of the time it is falling t, assuming air resistance is negligible. If an object dropped for t=1.5 seconds travels D=36 ft, how long will it take an object that is dropped to travel 120 feet? Show ALL work!

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Using model D=kt2, k=16 and t=2.74 seconds when D=120 ft
6.       The weight of a body W varies inversely as the square of its distance d from the center of the earth. A man weighs W=200 pounds on the face of the earth, which is d=4000 miles from the center. How much will the man weigh 1000 miles up from the surface, which is 5000 miles from the center of the earth? Show All Work!

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Using model W=k/d2, k=3200000000 and W=128 pounds when d=5000 mi
6.       The weight of a body W varies inversely as the square of its distance d from the center of the earth. A man weighs W=180 pounds on the face of the earth, which is d=4000 miles from the center. How much will the man weigh 1000 miles up, which is 5000 miles from the center of the earth? Show All Work!

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Using model W=k/d2, k=2880000000 and W=115.2 pounds when d=5000 mi
7.       If f(x) is as defined below, find and simplify f(t + 1).

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(t+1) / (t-1)
7.       If f(x) is as defined below, find and simplify f(t - 1).

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(t-1) / (t+1)
7.       If f(x) is as defined below, find and simplify f(t + 1).

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(t+2) / t
7.       If f(x) is as defined below, find and simplify f(t - 1).

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(t-2) / t
8.       State fhe domain of the function f(x) given below.

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All Real Numbers Except x=1 or x=-1
8.       State fhe domain of the function f(x) given below.

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All real numbers such that x is greater than or equal to 1.
8.       State fhe domain of the function f(x) given below.

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All Real Numbers
8.       State fhe domain of the function f(x) given below.

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All Real Numbers Such That x is Greater Than or Equal to 0.
9.       A rectangular floor has a length that is 4 feet more than its width. The area of the floor is 22 square feet. Find the length and width of this floor. Round to nearest hundredth of a foot. Note that area equals length times width. Show all work!

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W=3.10 and L=7.10
9.       A door has a length that is 4 feet more than its width. The area of the door is 14 square feet. Find the length and width of this door. Round to nearest hundredth of a foot. Note that area equals length times width. Show all work!

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W = 2.24 and L = 6.24
9.       A rectangular floor has a width that is 4 feet less than its length. The area of the floor is 24 square feet. Find the length and width of this floor. Round to nearest hundredth of a foot. Note that area equals length times width. Show all work!

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W=3.29 and L=7.29
9.       A door has a width that is 4 feet less than its length. The area of the door is 15 square feet. Find the length and width of this door. Round to nearest hundredth of a foot. Note that area equals length times width. Show all work!

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W = 2.36 and L = 6.36
10.       Find the equation of the line passing through the points shown below. Show All Work!

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y = 2x - 3
10.       Find the equation of the line passing through the points shown below. Show All Work!

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y = 2x - 1
10.       Find the equation of the line passing through the points shown below. Show All Work!

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y = 2x - 2
10.       Find the equation of the line passing through the points shown below. Show All Work!

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y = 3x - 1
11.       A ball is thrown up into the air and its height y is given by the function y = -16t2 + 48t + 8 where t is time in seconds. At what value of t does the ball obtain a maximum height? Show All Work!

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t = 1.5 sec
11.       A ball is thrown up into the air and its height y is given by the function y = -16t2 + 48t + 8 where t is time in seconds. At what value of t does the ball hit the ground? Round to the nearest hundredth of a second. Show All Work!

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t = 3.16 seconds
11.       A ball is thrown up into the air and its height y is given by the function y = -16t2 + 56t + 8 where t is time in seconds. At what value of t does the ball hit the ground? Round to the nearest hundredth of a second. Show All Work!

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t = 3.64 seconds
11.       A ball is thrown up into the air and its height y is given by the function y = -16t2 + 56t + 8 where t is time in seconds. At what value of t does the ball obtain a maximum height? Show All Work!

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t = 1.75 seconds
12.       Sketch the graph of f(x) = -2(x - 1)2. Show the vertex and at least two other points.

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12.       Sketch the graph of f(x) = 2(x + 1)2 - 3. Show the vertex and at least two other points.

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12.       Sketch the graph of f(x) = 2(x - 4)2 + 1. Show the vertex and at least two other points.

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12.       Sketch the graph of f(x) = -2(x + 4)2. Show the vertex and at least two other points.

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13.       The graph of the absolute value function is given below along with a shifted graph. Identify the equation of the shifted graph.

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13.       The graph of the absolute value function is given below along with a shifted graph. Identify the equation of the shifted graph.

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13.       The graph of the absolute value function is given below along with a shifted graph. Identify the equation of the shifted graph.

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13.       The graph of the absolute value function is given below along with a shifted graph. Identify the equation of the shifted graph.

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14.       Given f(x) = 2x2 + 1, evaluate and simplify the expression shown below. Show Work!

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4x + 2h
14.       Given f(x) = 3x2 + 2, evaluate and simplify the expression shown below. Show Work!

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6x + 3h
14.       Given f(x) = 2x2 + 2, evaluate and simplify the expression shown below. Show Work!

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4x + 2h
14.       Given f(x) = 3x2 + 1, evaluate and simplify the expression shown below. Show Work!

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6x + 3h
15.       Given f(x) = 3x + 2, find the inverse function f-1(x). Then show that (f o g)(x) = x and (g o f)(x) = x.

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Inverse is f-1(x) = (x-2)/3 and (f o f-1)(x) = 3[(x-2)/3) + 2 = x - 2 + 2 = x and (f-1 o f)(x) = (3x+2 - 2)/3 = 3x/x = x
15.       Given f(x) = 2x + 3, find the inverse function f-1(x). Then show that (f o g)(x) = x and (g o f)(x) = x.

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Inverse is f-1(x) = (x-3)/2 and (f o f-1)(x) = 2[(x-3)/2) + 3 = x - 3 + 3 = x and (f-1 o f)(x) = (2x + 3 - 3)/2 = 2x/x = x
15.       Given f(x) = 4x + 2, find the inverse function f-1(x). Then show that (f o g)(x) = x and (g o f)(x) = x.

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Inverse is f-1(x) = (x-2)/4 and (f o f-1)(x) = 4[(x-2)/4) + 2 = x - 2 + 2 = x and (f-1 o f)(x) = (4x+2 - 2)/4 = 4x/x = x
15.       Given f(x) = 2x + 4, find the inverse function f-1(x). Then show that (f o g)(x) = x and (g o f)(x) = x.

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Inverse is f-1(x) = (x-4)/2 and (f o f-1)(x) = 2[(x-4)/2) + 4 = x - 4 + 4 = x and (f-1 o f)(x) = (2x+4 - 4)/2 = 2x/x = x
16.       Use polynomial division to show that (x - 2) is a factor of f(x) = 2x3 - 2x2 - 3x - 2. Then use this result to write f(x) as the product of (x - 2) times a quadratic. Show All Work!

Answer Below:

(x - 2)(2x2 + 2x + 1)
16.       Use polynomial division to show that (x - 3) is a factor of f(x) = 2x3 - 3x2 - 3x - 18. Then use this result to write f(x) as the product of (x - 3) times a quadratic. Show All Work!

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(x - 3)(2x2 + 3x + 6)
16.       Use polynomial division to show that (x - 1) is a factor of f(x) = 2x3 - 2x2 - 3x + 3. Then use this result to write f(x) as the product of (x - 1) times a quadratic. Show All Work!

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(x - 1)(2x2 - 3)
16.       Use polynomial division to show that (x - 2) is a factor of f(x) = 3x3 - 2x2 - 5x - 6. Then use this result to write f(x) as the product of (x - 2) times a quadratic. Show All Work!

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(x - 2)(3x2 + 4x + 3)
17.       Graph the rational function shown below. Indicate ALL asymptotes of the graph and show at least 4 points on the graph.

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17.       Graph the rational function shown below. Indicate ALL asymptotes of the graph and show at least 4 points on the graph.

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17.       Graph the rational function shown below. Indicate ALL asymptotes of the graph and show at least 4 points on the graph.

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17.       Graph the rational function shown below. Indicate ALL asymptotes of the graph and show at least 4 points on the graph.

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18.       Rewrite LOG48 = 1.5 in exponential form and rewrite 3x = 2 in logarithmic form.

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41.5 = 8 and LOG32 = x
18.       Rewrite LOG927 = 1.5 in exponential form and rewrite 2x = 3 in logarithmic form.

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91.5 = 27 and LOG23 = x
18.       Rewrite LOG1664 = 1.5 in exponential form and rewrite x2 = 3 in logarithmic form.

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161.5 = 64 and LOGx3 = 2
18.       Rewrite LOG432 = 2.5 in exponential form and rewrite x3 = 2 in logarithmic form.

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42.5 = 32 and LOGx2 = 3
19.       Write the expression below as a sum or difference of logs or multiples of logs.

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½ [LOG10x + LOG10(x + 2)] OR ½ LOG10x + ½ LOG10(x + 2)
19.       Write the expression below as a sum or difference of logs or multiples of logs.

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3LN x + LN y
19.       Write the expression below as a sum or difference of logs or multiples of logs.

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3LN x - LN y
19.       Write the expression below as a sum or difference of logs or multiples of logs.

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½ LN x - ½ LN y
20.       Write 2LN x + ½ LN(x + 4) as a single logarithm by applying properties of logarithms.

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20.       Write ½ LN x + 2LN (x + 4) as a single logarithm by applying properties of logarithms.

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20.       Write 2LN x - ½ LN(x + 4) as a single logarithm by applying properties of logarithms.

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20.       Write ½ LN (x + 4) - 2LN x as a single logarithm by applying properties of logarithms.

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21.       Solve the equation LOG10x + LOG10(x + 3) = 1. Show All Work!

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x = 2 ONLY (throw out x = - 5)
21.       Solve the equation LOG10(x + 2) + LOG10(x + 3) = 1. Round to 3 decimal places. Show All Work!

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x = 0.702 ONLY (throw out other solution)
21.       Solve the equation LOG10(x + 5) + LOG10x = 2. Round to 3 decimal places. Show All Work!

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x = 5 ONLY (throw out x = - 20)
21.       Solve the equation LOG10(x + 3) + LOG10(x + 1) = 1. Round to 3 decimal places. Show All Work!

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x = 1.317 ONLY (throw out other answer)
22.       A population of bacteria is growing exponentially according to the function N(t) = 200ekt where N(t) is the number of bacteria at time t hours. If N(t)= 400 when t = 5 hours, what will the population be when t = 12 hours? Round final answer to nearest whole number. Show All Work!

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k = 0.1386, N = 1055 when t=12 hours
22.       A population of bacteria is growing exponentially according to the function N(t) = 200ekt where N(t) is the number of bacteria at time t hours. If N(t)= 500 when t = 4 hours, what will the population be when t = 12 hours? Round final answer to nearest whole number. Show All Work!

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k = 0.229, N = 3122 when t=12 hours
22.       A population of bacteria is growing exponentially according to the function N(t) = 400ekt where N(t) is the number of bacteria at time t hours. If N(t)= 800 when t = 5 hours, what will the population be when t = 12 hours? Round final answer to nearest whole number. Show All Work!

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k = 0.1386, N = 2110 when t=12 hours
22.       A population of bacteria is growing exponentially according to the function N(t) = 500ekt where N(t) is the number of bacteria at time t hours. If N(t)= 900 when t = 5 hours, what will the population be when t = 12 hours? Round final answer to nearest whole number. Show All Work!

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k = 0.1176, N = 1621 when t=10 hours
23.       Solve 2e3x + 4 = 7. Round final answer to 3 decimal places. Show All Work!

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x = 0.135
23.       Solve 3e2x + 6 = 10. Round final answer to 3 decimal places. Show All Work!

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x = 0.144
23.       Solve 2e2x + 5 = 9. Round final answer to 3 decimal places. Show All Work!

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x = 0.346
23.       Solve 2e4x + 3 = 4. Round final answer to 3 decimal places. Show All Work!

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x = - 0.173
24.       A person selling widgets sells 100 widgets/day and makes $40 net profit for the day. If they increase their number of widgets sold per day to 120 widgets/day they make $65 in net profit per day. Assuming there is a linear relationship between number of widgets sold per day (x) and net profit per day (y), find the following: A) A linear model that relates net daily profit (y) to widgets sold per day (x). Write as a linear equation. Hint: This amounts to finding the equation of a line. B) The amount of net profit made daily (or lost) if there ZERO widgets are sold each day. Hint: Plug into the formula you found in Part A. Show All Work!

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y = 1.25x - 85, $85 is lost per day if x=0 widgets are sold
24.       A person selling widgets sells 100 widgets/day and makes $40 net profit for the day. If they increase their number of widgets sold per day to 120 widgets/day they make $75 in net profit per day. Assuming there is a linear relationship between number of widgets sold per day (x) and net profit per day (y), find the following: A) A linear model that relates net daily profit (y) to widgets sold per day (x). Write as a linear equation. Hint: This amounts to finding the equation of a line. B) The amount of net profit made daily (or lost) if there ZERO widgets are sold each day. Hint: Plug into the formula you found in Part A. Show All Work!

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y = 1.75x - 135, $135 is lost per day if x=0 widgets are sold
24.       A person selling widgets sells 100 widgets/day and makes $40 net profit for the day. If they increase their number of widgets sold per day to 120 widgets/day they make $70 in net profit per day. Assuming there is a linear relationship between number of widgets sold per day (x) and net profit per day (y), find the following: A) A linear model that relates net daily profit (y) to widgets sold per day (x). Write as a linear equation. Hint: This amounts to finding the equation of a line. B) The amount of net profit made daily (or lost) if there ZERO widgets are sold each day. Hint: Plug into the formula you found in Part A. Show All Work!

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y = 1.5x - 110, $110 is lost per day if x=0 widgets are sold
24.       A person selling widgets sells 100 widgets/day and makes $40 net profit for the day. If they increase their number of widgets sold per day to 120 widgets/day they make $95 in net profit per day. Assuming there is a linear relationship between number of widgets sold per day (x) and net profit per day (y), find the following: A) A linear model that relates net daily profit (y) to widgets sold per day (x). Write as a linear equation. Hint: This amounts to finding the equation of a line. B) The amount of net profit made daily (or lost) if there ZERO widgets are sold each day. Hint: Plug into the formula you found in Part A. Show All Work!

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y = 2.75x - 235, $235 is lost per day if x=0 widgets are sold
25.       X units are sold per day at a profit of $2 per day, not including daily overhead costs. Daily overhead costs are $90. Find a function f(x) that gives the net profit per day that takes into account the overhead costs. Also, what is the minimum number of units that must be sold each day just to make a net profit of $10 per day? Show ALL work!

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f(x) = 2x - 90, x=50 results in $10 profit per day
25.       X units are sold per day at a profit of $3 per day, not including daily overhead costs. Daily overhead costs are $90. Find a function f(x) that gives the net profit per day that takes into account the overhead costs. Also, what is the minimum number of units that must be sold each day just to make a net profit of $30 per day? Show ALL work!

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f(x) = 3x - 90, x=40 results in $30 profit per day
25.       X units are sold per day at a profit of $2 per day, not including daily overhead costs. Daily overhead costs are $80. Find a function f(x) that gives the net profit per day that takes into account the overhead costs. Also, what is the minimum number of units that must be sold each day to make a net profit of $100 per day? Show ALL work!

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f(x) = 2x - 80, x=90 results in $100 profit per day
25.       X units are sold per day at a profit of $5 per day, not including daily overhead costs. Daily overhead costs are $90. Find a function f(x) that gives the net profit per day that takes into account the overhead costs. Also, what is the minimum number of units that must be sold each day just to break even? Show ALL work!

Answer Below:

f(x) = 5x - 90, x=18 results in break even

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