12 multiple-choice questions, progressively harder.
Two positive numbers add up to 202020. What is their greatest possible product?
Solution
Correct answer: B
Name one number xxx; the other is 20−x20 - x20−x. Their product is a quadratic.
P=x(20−x)=20x−x2P = x(20 - x) = 20x - x^2P=x(20−x)=20x−x2
With a=−1<0a = -1 < 0a=−1<0 the vertex is a maximum, at x=−202(−1)=10x = -\tfrac{20}{2(-1)} = 10x=−2(−1)20=10. Both numbers are 101010, so the greatest product is P=10⋅10=100P = 10 \cdot 10 = 100P=10⋅10=100. The number 101010 is the input; the product 100100100 is the answer.
A rectangle has a perimeter of 242424. What width gives it the greatest area?
Correct answer: D
Let the width be xxx. Two widths and two lengths total 242424, so the length is 12−x12 - x12−x, and the area is A=x(12−x)A = x(12 - x)A=x(12−x).
x=−b2a=−122(−1)=6x = -\frac{b}{2a} = -\frac{12}{2(-1)} = 6x=−2ab=−2(−1)12=6
The width that maximizes area is 666 (the rectangle is a 666 by 666 square).
A rocket's height in feet is h=−16t2+80th = -16t^2 + 80th=−16t2+80t. When does it reach its greatest height?
Correct answer: A
The height is a downward parabola, so the peak is at the vertex time t=−b2at = -\tfrac{b}{2a}t=−2ab with a=−16a = -16a=−16, b=80b = 80b=80.
t=−802(−16)=−80−32=2.5t = -\frac{80}{2(-16)} = -\frac{80}{-32} = 2.5t=−2(−16)80=−−3280=2.5
The rocket is highest at t=2.5t = 2.5t=2.5 seconds. The value 100100100 feet is the height there, which answers a different question.
A rectangle has a perimeter of 242424. What is its greatest possible area?
Correct answer: C
With width xxx the length is 12−x12 - x12−x, so A=x(12−x)A = x(12 - x)A=x(12−x), largest at the vertex x=6x = 6x=6. Substitute back for the area.
A=6(12−6)=6⋅6=36A = 6(12 - 6) = 6 \cdot 6 = 36A=6(12−6)=6⋅6=36
The greatest area is 363636, from a 666 by 666 square.
A machine's running cost in dollars is C=x2−12x+50C = x^2 - 12x + 50C=x2−12x+50, where xxx is the speed setting. What is the least possible cost?
Since a=1>0a = 1 > 0a=1>0 the cost has a minimum at the vertex, whose input is x=−−122=6x = -\tfrac{-12}{2} = 6x=−2−12=6. Substitute back for the cost.
C=(6)2−12(6)+50=36−72+50=14C = (6)^2 - 12(6) + 50 = 36 - 72 + 50 = 14C=(6)2−12(6)+50=36−72+50=14
The least cost is 141414 dollars, reached at the setting x=6x = 6x=6.
For that stall, with revenue R=x(100−x)R = x(100 - x)R=x(100−x) dollars, what is the greatest revenue?
The revenue peaks at the price x=50x = 50x=50. Substitute it back to get the revenue value.
R=50(100−50)=50⋅50=2500R = 50(100 - 50) = 50 \cdot 50 = 2500R=50(100−50)=50⋅50=2500
The greatest revenue is 250025002500 dollars. The price 505050 dollars is where it happens, a separate number.
Two positive numbers add up to 505050. Which pair gives the greatest product?
With one number xxx, the product P=x(50−x)P = x(50 - x)P=x(50−x) is a downward parabola, largest at the vertex x=25x = 25x=25.
x=−502(−1)=25 ⇒ other number 50−25=25x = -\frac{50}{2(-1)} = 25 \;\Rightarrow\; \text{other number } 50 - 25 = 25x=−2(−1)50=25⇒other number 50−25=25
The equal pair 252525 and 252525 gives the greatest product. A fixed sum is best split evenly.
A ball's height in feet is h=−16t2+64t+3h = -16t^2 + 64t + 3h=−16t2+64t+3. What is its maximum height?
The peak is at the vertex time t=−642(−16)=2t = -\tfrac{64}{2(-16)} = 2t=−2(−16)64=2 seconds. Substitute back for the height.
h=−16(2)2+64(2)+3=−64+128+3=67h = -16(2)^2 + 64(2) + 3 = -64 + 128 + 3 = 67h=−16(2)2+64(2)+3=−64+128+3=67
The maximum height is 676767 feet. The time 222 seconds is when it happens, not how high.
A rectangle has a perimeter of 404040, so its area is A=x(20−x)A = x(20 - x)A=x(20−x). What is its greatest area?
The area peaks at the vertex x=−202(−1)=10x = -\tfrac{20}{2(-1)} = 10x=−2(−1)20=10. Substitute back.
A=10(20−10)=10⋅10=100A = 10(20 - 10) = 10 \cdot 10 = 100A=10(20−10)=10⋅10=100
The greatest area is 100100100, from a 101010 by 101010 square.
For that shop, with R=p(60−p)R = p(60 - p)R=p(60−p) dollars, what is the greatest revenue?
Revenue peaks at the price p=30p = 30p=30. Substitute back to get the revenue value.
R=30(60−30)=30⋅30=900R = 30(60 - 30) = 30 \cdot 30 = 900R=30(60−30)=30⋅30=900
The greatest revenue is 900900900 dollars, at a price of 303030 dollars.
A rectangle has a perimeter of 282828, so its area is A=x(14−x)A = x(14 - x)A=x(14−x). What is its greatest area?
The area peaks at the vertex x=−142(−1)=7x = -\tfrac{14}{2(-1)} = 7x=−2(−1)14=7. Substitute back.
A=7(14−7)=7⋅7=49A = 7(14 - 7) = 7 \cdot 7 = 49A=7(14−7)=7⋅7=49
The greatest area is 494949, from a 777 by 777 square.
A stone's height in feet is h=−16t2+48t+5h = -16t^2 + 48t + 5h=−16t2+48t+5. When does it reach its maximum height?
The peak is at the vertex time t=−b2at = -\tfrac{b}{2a}t=−2ab with a=−16a = -16a=−16, b=48b = 48b=48.
t=−482(−16)=−48−32=1.5t = -\frac{48}{2(-16)} = -\frac{48}{-32} = 1.5t=−2(−16)48=−−3248=1.5
The stone is highest at t=1.5t = 1.5t=1.5 seconds. The value 414141 feet is the height there, not the time.
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