Multiple Choice
Your café sells lattes for \$4 each to 100 customers per day. For every \$1 increase in price, you would lose 20 customers. Find the price that maximizes revenue. Hint: The # of items sold is based on the number of customers.
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6. Graphical Applications of Derivatives
Applied Optimization
Multiple Choice
A poster is set to have a total area of 1150 cm2, with 2-cm margins on the sides and the top, and a 3-cm margin at the bottom. What dimensions will maximize the printed area?
A
Width: 22.5 cm, Height: 51.2 cm
B
Width: 35.6 cm, Height: 32.3 cm
C
Width = 30.3 cm, Height = 37.9 cm
D
Width: 28.1 cm, Height: 41.0 cm
Verified step by step guidance1
Step 1: Define the variables for the dimensions of the poster. Let the width of the poster be 'W' and the height of the poster be 'H'. The total area of the poster is given as 1150 cm².
Step 2: Account for the margins. The printed area is reduced by 2 cm on each side and the top, and 3 cm at the bottom. Therefore, the dimensions of the printed area are (W - 4) cm for the width and (H - 5) cm for the height.
Step 3: Write the equation for the printed area. The printed area is given by the product of the adjusted width and height: Printed Area = (W - 4) * (H - 5).
Step 4: Use the constraint for the total area of the poster. The total area is given as 1150 cm², so W * H = 1150. Solve this equation for one variable (e.g., H = 1150 / W) and substitute it into the printed area equation.
Step 5: Maximize the printed area. Use calculus to find the maximum value of the printed area. Take the derivative of the printed area equation with respect to W, set it equal to zero, and solve for W. Then, use the constraint equation to find the corresponding value of H.
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