Question

you want to enlarge a photo to make a poster. the poster will have the same length width ratio as the photo. the photo is 7 inches by 5 inches. you want the poster to have an area that is at least 250% as large as the are of the photo. find the minimum dimensions of the poster. round the dimensions to the nearest tenth of an ince

Answer 1

Task:

You want to enlarge a photo to make a poster. The poster will have the same length-width ratio as the photo. The photo is 7 inches by 5 inches. You want the poster to have an area that is at least $250\%$ as large as there are of the photo. Find the minimum dimensions of the poster, round the dimensions to the nearest tenth of an inch.

Solution:

Area of the photo:

7 inches x 5 inches=35 square inches

The poster is $250\%$ larger, area of the poster:

2.5 x 35 square inches=70 square inches

Area of the poster equals the multiplication of its dimensions $a$ and $b$ :

a inches x b inches=70 square inches

$a = 70 / b$

To find the minimum dimensions we should find the minimum of the function:

$$y = a + b = \frac {7 0}{b} + b, b > 0, y ^ {\prime} = 1 - \frac {7 0}{b ^ {2}}$$

In the interval $b > 0$ there is a minimum:

$$1 - \frac {7 0}{b ^ {2}} = 0, b > 0$$ $$b ^ {2} = 7 0$$ $$b = \sqrt {7 0} = 8. 4 \mathrm {i n c h}$$ $$a = \frac {7 0}{b} = \frac {7 0}{\sqrt {7 0}} = 8. 4 \mathrm {i n c h}$$

Control: $a \cdot b = 70.56$ square inches $\delta = \frac{a \cdot b - 70}{70} = 0.008$

Answer: $b = 8.4$ inch, $a = 8.4$ inch