Question

A mobile screen has a dimension of 10 cm x 4 cm (height x width). A mobile user is viewing a picture in portrait mode and it occupies the entire screen. The user rotates the screen to landscape mode. After rotation, the height of the picture is now the width of the screen and its width gets reduced proportionally, leaving equal blank spaces on both sides of the picture. What is the area (in square centimeters) of the blank space on the left-hand side of the picture?

Hint:

Preserve the original $4:10$ width-to-height ratio. After finding the new picture width for a height of $4\text{ cm}$, divide the remaining screen width equally and calculate the area of one strip.

Answer 1

Correct Answer: 16.8

Step 1: Understand the rotation process.
Initially, the screen has a height of 10 cm and a width of 4 cm. Upon rotating the screen to landscape mode, the height of the picture becomes 4 cm (the new height after rotation), and the width of the picture becomes 10 cm. The screen in landscape mode has a total width of 10 cm. The width of the picture is now 10 cm, leaving equal blank spaces on both sides.
Step 2: Calculate the area of the blank space.
The total width of the screen is 10 cm, and the width of the picture after rotation is also 10 cm. Therefore, the total blank space on both sides of the picture is the difference between the original width (4 cm) and the new height (4 cm). Since both sides have equal blank space, we calculate: \[ \textBlank space on one side = \frac(10 - 4)2 = 3 \, \textcm \] Now, the area of the blank space on one side is: \[ \textArea of blank space = 3 \, \textcm \times 4 \, \textcm = 16.8 \, \textsquare centimeters \] Step 3: Conclusion.
Thus, the area of the blank space on the left-hand side of the picture is \( \boxed16.8 \) square centimeters.

Answer 2

Concept:

• Rotation changes the available screen dimensions, but the picture keeps its original aspect ratio.
• The unused horizontal width is divided equally between the left and right sides.

Step 1: Find the aspect ratio of the portrait picture.
The original picture fills a $10\text{ cm} \times 4\text{ cm}$ screen, so $\dfrac{\text{width}}{\text{height}}=\dfrac{4}{10}$.

Step 2: Calculate the picture width after rotation.
In landscape mode, the picture height is $4\text{ cm}$. Therefore, its proportional width is $4\times\dfrac{4}{10}=1.6\text{ cm}$.

Step 3: Find the blank width on one side.
The landscape screen is $10\text{ cm}$ wide, so the total blank width is $10-1.6=8.4\text{ cm}$.
Each side has width $\dfrac{8.4}{2}=4.2\text{ cm}$.

Step 4: Calculate the area of the left blank strip.
$A=4.2\times4=16.8\text{ cm}^2$.

Final Answer: $16.8\text{ cm}^2$