Question

A rectangular piece of cardboard measure 17cm by 14cm. Strips of equal width are cut off one side and one End. The area of the remaining piece is 108cm² find the width of the strips removed

Answer 1

Let the width of the strips cut off from one side and one end be \( x \) cm.

After cutting off the strips, the dimensions of the remaining piece are:
- Length: \( 17 - x \) cm
- Width: \( 14 - x \) cm

The area of the remaining piece is given by:

\[
(17 - x)(14 - x) = 108
\]

Now, let's expand this equation:

\[
238 - 17x - 14x + x^2 = 108
\]

Simplifying it, we obtain:

\[
x^2 - 31x + 238 - 108 = 0
\]

\[
x^2 - 31x + 130 = 0
\]

Next, we can solve this quadratic equation using the quadratic formula:

\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\]

Here, \( a = 1 \), \( b = -31 \), and \( c = 130 \).

Calculating the discriminant:

\[
b^2 - 4ac = (-31)^2 - 4 \cdot 1 \cdot 130 = 961 - 520 = 441
\]

Now substituting back into the quadratic formula:

\[
x = \frac{{31 \pm \sqrt{441}}}{2} = \frac{{31 \pm 21}}{2}
\]

Calculating the two possible values for \( x \):

1. \( x = \frac{{31 + 21}}{2} = \frac{52}{2} = 26 \)
2. \( x = \frac{{31 - 21}}{2} = \frac{10}{2} = 5 \)

Since \( x = 26 \) cm is not possible (as it exceeds the small side of the cardboard), we have:

\[
x = 5 \text{ cm}
\]

Thus, the width of the strips removed is \( \boxed{5} \) cm.