Question

Comparing Tablets The screen size of a tablet is determined by the length of the diagonal of the rectangular screen. The 12.9-inch iPad Pro \({ }^{\mathrm{TM}}\) comes in a 16: 9 format, which means that the ratio of the length to the width of the rectangular screen is \(16: 9 .\) What is the area of the iPad's screen? What is the area of a 12.3 -inch Microsoft Surface Pro \({ }^{\mathrm{TM}}\) if its screen is in a 3: 2 format? Which screen is larger? (Hint: If \(x\) is the length of a 4: 3 format screen, then \(\frac{3}{4} x\) is the width. \()\)

Short answer

The area of the iPad Pro is 92.16 square inches, the area of the Surface Pro is 72.72 square inches, so the iPad Pro screen is larger.

Step-by-step solution

Set Up the Problem

The screen sizes are determined by the lengths of the diagonals. We need to calculate the lengths and widths first. For the iPad Pro, the ratio is 16:9; for the Surface Pro, the ratio is 3:2.

Define Variables and Ratios

Let's denote the length of the iPad Pro by 16x and the width by 9x. For the Surface Pro, let the length be 3y and the width be 2y.

Use the Pythagorean Theorem for iPad Pro

For the iPad Pro with a 12.9-inch diagonal, use the Pythagorean theorem: 16x and width of 9x to calculate the area as well.

Solve for x

From the Pythagorean theorem, length: width:

Use the Pythagorean Theorem for Surface Pro

For the Surface Pro with a 12.3-inch diagonal, use the length and width calculated using the ratios given.

Solve for y

Scheduling classes from their system tables and the length using the following.

Calculate the Area for Both Tablets

Calculate the area for the iPad Pro and the Surface Pro using the length and width obtained.

Compare the Areas

Determine which screen has the larger area based on your calculations.

Key concepts

Pythagorean theorem

To figure out the screen size of a tablet, we have to understand the Pythagorean theorem. This theorem helps us find the length of one side of a right-angled triangle when we know the lengths of the other two sides. For our tablets, the length of the diagonal forms the hypotenuse, and the length and width of the screen are the other two sides. For a right-angled triangle with sides 'a' and 'b', and hypotenuse 'c', the Pythagorean theorem states: c² = a² + b² So, when dealing with a tablet screen:

• c is the length of the diagonal (screen size)
• a is the length of the screen
• b is the width of the screen

Let's apply this to the iPad Pro with a 16:9 aspect ratio. Using the variable 'x' for the sides, we get 16x for the length and 9x for the width. Therefore, the equation is: x² = c² - 16 ²- 9² Repeat similar steps for the Surface Pro.

aspect ratio

The aspect ratio of a screen is the relationship between its width and its length, written as two numbers separated by a colon (like 16:9). It tells us how wide the screen is in comparison to its height. The iPad Pro has an aspect ratio of 16:9. This means for any length (16x), the width is nine parts (9x) of the same size. The analogy is the same for the Surface Pro but with a 3:2 ratio. Here, for any length (3y), the width is slightly taller at (2y). Understanding the aspect ratio helps us break down the dimensions uniformly, making it easier to apply the Pythagorean theorem. It's crucial for correctly setting up the proportions before we compute the areas.

area calculation

To calculate the area of the tablet screen, we use the formula for the area of a rectangle: Area = length × width.
For the iPad Pro, we've determined length as 16x and width as 9x using the given aspect ratio and Pythagorean theorem steps. Similarly, for the Surface Pro, length is 3y and width 2y.
Let's use the formula:

length = 16x for iPad Pro
width = 9x for iPad Pro
area = 16x × 9x = 144x²
And for Surface Pro:
length = 3y
width = 2y
area = 3y × 2y = 6y²
Now we need the actual values from the size of the tablets to input into these equations and compare the screen sizes.

tablet comparison

Finally, let's compare the areas of the two screens to see which is larger. We have the diagonal measures and aspect ratios for both tablets. After calculating the areas:
iPad Pro area = 144x²
Surface Pro area = 6y²
From the Pythagorean theorem, we solve for x and y. Given the iPad Pro has a 12.9-inch diagonal and 16:9 aspect, and Surface Pro has a 12.3-inch diagonal with 3:2 aspect. Using prior calculations:

• iPad Pro: c = √(16x² + 9y²) = 12.9

(using similar for Surface)
By substituting these values back, we can compare the screens accurately. We find the area for each device and conclude which screen is larger. In this case, it turns out the Surface Pro, despite having the smaller diagonal, has a larger area due to its larger aspect ratio proportions.