Question

A playing card has a perimeter of 34 centimeters. Its area is 70 square centimeters. What are the dimensions of the playing card?

centimeters by
centimeters

Answer 1

Let the dimensions of the playing card be \( x \) centimeters (length) and \( y \) centimeters (width).

We know the following two equations based on the information given:

1. The perimeter of the card is given by:
\[
2x + 2y = 34
\]
This can be simplified to:
\[
x + y = 17 \quad \text{(Equation 1)}
\]

2. The area of the card is given by:
\[
x \cdot y = 70 \quad \text{(Equation 2)}
\]

To solve for \( x \) and \( y \), we can use Equation 1 to express \( y \) in terms of \( x \):
\[
y = 17 - x
\]

Now, substitute this expression for \( y \) into Equation 2:
\[
x(17 - x) = 70
\]
This simplifies to:
\[
17x - x^2 = 70
\]
Rearranging gives:
\[
x^2 - 17x + 70 = 0
\]

Next, we can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \( a = 1 \), \( b = -17 \), and \( c = 70 \):
\[
x = \frac{17 \pm \sqrt{(-17)^2 - 4 \cdot 1 \cdot 70}}{2 \cdot 1}
\]
Calculating the discriminant:
\[
x = \frac{17 \pm \sqrt{289 - 280}}{2}
\]
\[
x = \frac{17 \pm \sqrt{9}}{2}
\]
\[
x = \frac{17 \pm 3}{2}
\]

This gives us two possible solutions for \( x \):
1. \( x = \frac{20}{2} = 10 \)
2. \( x = \frac{14}{2} = 7 \)

Now we find \( y \) for each case:

**Case 1: If \( x = 10 \)**
\[
y = 17 - 10 = 7
\]

**Case 2: If \( x = 7 \)**
\[
y = 17 - 7 = 10
\]

Thus, the dimensions of the playing card are \( 10 \) centimeters by \( 7 \) centimeters.

Therefore, the final answer is:

10 centimeters by 7 centimeters.