Question

adam wraps the top edge of the gift box shown with gold ribbon. the box has a volume of 296 cm³. the top and bottom sides of the box are square. if adam has 24 1/4 cm of gold ribbon, does he have enough to decorate the top of the box? the amount of gold ribbon needed to wrap the top edge of the box is □. therefore, adam □ enough ribbon to wrap the top of the box. (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Find the side length of the square base

Let the side length of the square base be \(x\) cm and the height be \(h\) cm. The volume formula of a rectangular prism is \(V=x\times x\times h=x^{2}h\). Given \(V = 296\) \(cm^{3}\). Also, assume \(h = 4\) (by trial - and - error or factoring \(296=2\times2\times2\times37\), and if we assume a reasonable non - fractional value for \(h\) related to common ribbon - wrapping scenarios, we can check \(h = 4\)). Then \(x^{2}=\frac{V}{h}=\frac{296}{4}=74\), so \(x=\sqrt{74}\approx8.60\) cm.

Step2: Calculate the perimeter of the square top

The perimeter of a square is \(P = 4x\). Substituting \(x=\sqrt{74}\), we get \(P = 4\sqrt{74}\approx4\times8.60 = 34.40\) cm.

Step3: Compare the ribbon length

Adam has \(24.25\) cm of ribbon. Since \(34.40>24.25\)

Answer:

The amount of gold ribbon needed to wrap the top edge of the box is \(34.40\) cm. Therefore, Adam does not have enough ribbon to wrap the top of the box.