Rafael wants to use 30 centimeters of wood to frame a rectangular photo. What is the greatest area his photograph could have?
a)14 sqaure cm
b)36 square cm
c)54 square cm
d)56 square cm
P = 2L + 2W
The dimensions could be:
2 by 13
3 by 12
4 by 11
5 by 10
6 by 9
7 by 8
What is the largest area of these dimensions?
56
To find the greatest possible area of the photograph, we need to determine the dimensions of the rectangular frame that would use 30 cm of wood.
Let's assume the length of the frame is L cm and the width is W cm. To calculate the amount of wood needed for the frame, we use the formula: perimeter = 2L + 2W.
Given that the perimeter is 30 cm, we have the equation: 2L + 2W = 30.
However, we want to maximize the area, which is given by A = L * W.
To find the greatest possible area, we need to maximize A while also satisfying the equation for the perimeter.
We can solve the equation 2L + 2W = 30 for W and rewrite it as W = (30 - 2L) / 2.
Substituting this value of W in the equation for the area, we get A = L * (30 - 2L) / 2.
To find the maximum value of A, we can take the derivative of A with respect to L, set it equal to zero, and solve for L. However, since it's a multiple-choice question, we can try plugging in the values of L given for each option and see which one gives us the highest area.
Let's calculate the area for each option:
a) L = 7, W = (30 - 2 * 7) / 2 = 30 / 2 - 14 / 2 = 8. Area = 7 * 8 = 56 square cm.
b) L = 9, W = (30 - 2 * 9) / 2 = 30 / 2 - 18 / 2 = 6. Area = 9 * 6 = 54 square cm.
c) L = 10, W = (30 - 2 * 10) / 2 = 30 / 2 - 20 / 2 = 5. Area = 10 * 5 = 50 square cm.
d) L = 14, W = (30 - 2 * 14) / 2 = 30 / 2 - 28 / 2 = 1. Area = 14 * 1 = 14 square cm.
From the calculations, we see that the greatest area Rafael's photograph could have is 56 square cm. Therefore, the correct answer is option d) 56 square cm.