Question

3 observe les trois boîtes ci - dessous. pour les décorer, il faut :

• un ruban dans le sens de la longueur, de la largeur et de la hauteur placé comme sur la boîte cadeau ci - contre ;
• une boucle formée de 30 cm de ruban.

tu as un rouleau de 1,5 m de ruban à décorer.
(then there are three boxes with dimensions: green box: length 12 cm, width 10 cm, height 18 cm; purple box: length 30 cm, width 10 cm, height 10 cm; orange box: length 25 cm, width 11 cm, height 16 cm)

Explanation:

Step1: Analyze the ribbon for one box

For a rectangular box, the ribbon along length, width, height: we have 2 lengths, 2 widths, and 4 heights (from the diagram's ribbon placement: vertical parts are height, horizontal top/bottom are length and width). Wait, looking at the gift box diagram, the ribbon goes around: 2 times length, 2 times width, and 4 times height? Wait no, let's take the first green box: length \( l = 12 \) cm, width \( w = 10 \) cm, height \( h = 18 \) cm. The ribbon: 2*l + 2*w + 4*h? Wait no, maybe 2*(l + w) + 4*h? Wait, no, the standard ribbon for a box like that: when you tie it, you have two loops: one around length and height, one around width and height? Wait, the problem says "un ruban dans le sens de la longueur, de la largeur et de la hauteur placé comme sur la boîte cadeau ci - contre". So from the diagram, the red ribbon: vertical lines (height) and horizontal lines (length and width). So for a box with length \( l \), width \( w \), height \( h \), the ribbon length for the wrapping (excluding the bow) is \( 2l + 2w + 4h \)? Wait, no, let's calculate for the first box: \( l = 12 \), \( w = 10 \), \( h = 18 \). So 2*l (front and back length) + 2*w (left and right width) + 4*h (four vertical sides). So \( 2*12 + 2*10 + 4*18 = 24 + 20 + 72 = 116 \) cm. Then add the bow: 30 cm. So total per box: \( 116 + 30 = 146 \) cm? Wait, no, maybe I messed up. Wait the problem says "un ruban dans le sens de la longueur, de la largeur et de la hauteur" so maybe 2*(l + w + h)? No, that doesn't match. Wait let's check the second box: purple box, \( l = 30 \), \( w = 10 \), \( h = 10 \). Third box: orange box, \( l = 25 \), \( w = 11 \), \( h = 16 \). Wait the total ribbon per box is (2*l + 2*w + 4*h) + 30. Let's confirm with the first box: \( 2*12 + 2*10 + 4*18 = 24 + 20 + 72 = 116 \), plus 30 is 146. Second box: \( 2*30 + 2*10 + 4*10 = 60 + 20 + 40 = 120 \), plus 30 is 150. Third box: \( 2*25 + 2*11 + 4*16 = 50 + 22 + 64 = 136 \), plus 30 is 166? Wait no, that can't be. Wait maybe it's 2*(l + h) + 2*(w + h) = 2l + 2h + 2w + 2h = 2l + 2w + 4h. Yes, that's the same as before. Now, we have three boxes. Wait the total ribbon we have is 1.5 m = 150 cm? No, 1.5 m is 150 cm? Wait no, 1 m = 100 cm, so 1.5 m = 150 cm? Wait that can't be, because the first box alone would need 146 cm, plus the second and third would be way more. Wait maybe I misread the total ribbon. Wait the problem says "Tu as un rouleau de 1,5 m de ruban à décorer." 1.5 m is 150 cm? No, 1.5 m is 150 cm? Wait no, 1 meter is 100 centimeters, so 1.5 meters is 150 centimeters. But that seems too short. Wait maybe the ribbon length for the wrapping is 2*(l + w) + 4*h. Let's recalculate the first box: \( l = 12 \), \( w = 10 \), \( h = 18 \). 2*(12 + 10) + 4*18 = 2*22 + 72 = 44 + 72 = 116. Bow: 30. Total: 146. Second box: \( l = 30 \), \( w = 10 \), \( h = 10 \). 2*(30 + 10) + 4*10 = 80 + 40 = 120. Bow: 30. Total: 150. Third box: \( l = 25 \), \( w = 11 \), \( h = 16 \). 2*(25 + 11) + 4*16 = 2*36 + 64 = 72 + 64 = 136. Bow: 30. Total: 166. Now, sum of three boxes: 146 + 150 + 166 = 462 cm. But we have 1.5 m = 150 cm. That's a problem. Wait maybe I misinterpreted the ribbon placement. Maybe it's 2*(l + h) + 2*(w + h) = 2l + 2h + 2w + 2h = 2(l + w + 2h). Let's recalculate first box: 2*(12 + 10 + 2*18) = 2*(12 + 10 + 36) = 2*58 = 116. Same as before. Bow: 30. Total 146. Wait the total ribbon available is 1.5 m = 150 cm. So maybe the question is "Can we decorate all three boxes?" or "How much ribbon is needed?" Wait the original problem (in French) says "Observe les trois boîtes ci - dessous. Pour les décorer, il faut :... Tu as un rouleau de 1,5 m de ruban à décorer." Maybe the question is to calculate the total ribbon needed for the three boxes and compare with 1.5 m. Let's do that.

First, calculate ribbon for each box:

Box 1 (Green):

Length \( l_1 = 12 \) cm, Width \( w_1 = 10 \) cm, Height \( h_1 = 18 \) cm.

Ribbon for wrapping (excluding bow): \( 2l_1 + 2w_1 + 4h_1 \) (as per ribbon placement: 2 lengths, 2 widths, 4 heights)
\( 2*12 + 2*10 + 4*18 = 24 + 20 + 72 = 116 \) cm.

Add bow: \( 116 + 30 = 146 \) cm.

Box 2 (Purple):

Length \( l_2 = 30 \) cm, Width \( w_2 = 10 \) cm, Height \( h_2 = 10 \) cm.

Ribbon for wrapping: \( 2*30 + 2*10 + 4*10 = 60 + 20 + 40 = 120 \) cm.

Add bow: \( 120 + 30 = 150 \) cm.

Box 3 (Orange):

Length \( l_3 = 25 \) cm, Width \( w_3 = 11 \) cm, Height \( h_3 = 16 \) cm.

Ribbon for wrapping: \( 2*25 + 2*11 + 4*16 = 50 + 22 + 64 = 136 \) cm.

Add bow: \( 136 + 30 = 166 \) cm.

Total ribbon needed:

\( 146 + 150 + 166 = 462 \) cm.

Ribbon available: \( 1.5 \) m \( = 150 \) cm.

Since \( 462 > 150 \), we can't decorate all three. But maybe the question is to calculate the ribbon needed for one box? Wait the problem is not fully translated, but from the given, let's assume we need to calculate the total ribbon needed for the three boxes.

Wait, maybe I made a mistake in the ribbon formula. Let's look at the gift box diagram: the red ribbon has two horizontal loops (length and width) and four vertical loops (height). So horizontal: 2*(l + w), vertical: 4*h. So total wrapping ribbon: \( 2(l + w) + 4h \), then add bow (30 cm). So that's correct.

Now, convert 1.5 m to cm: \( 1.5 \times 100 = 150 \) cm.

Total needed: \( 146 + 150 + 166 = 462 \) cm.

Difference: \( 462 - 150 = 312 \) cm, but that seems too much. Wait maybe the ribbon for the wrapping is \( 2(l + h) + 2(w + h) = 2l + 2h + 2w + 2h = 2(l + w + 2h) \), which is the same as before.

Alternatively, maybe the ribbon is \( 4h + 2l + 2w \), which is what we used.

Wait, maybe the question is "How much ribbon is needed for one box?" Let's take the first box: \( 2*12 + 2*10 + 4*18 + 30 = 116 + 30 = 146 \) cm.

Second box: \( 2*30 + 2*10 + 4*10 + 30 = 120 + 30 = 150 \) cm.

Third box: \( 2*25 + 2*11 + 4*16 + 30 = 136 + 30 = 166 \) cm.

If the question is to find the total ribbon needed for the three boxes: \( 146 + 150 + 166 = 462 \) cm \( = 4.62 \) m, but we have 1.5 m, so we can't. But maybe the question is different. Since the original problem is in French, maybe the question is to calculate the length of ribbon needed for one box, or compare with 1.5 m.

Assuming the question is "Calculate the total length of ribbon needed to decorate the three boxes"

Step1: Calculate ribbon for Box 1

\( l = 12 \), \( w = 10 \), \( h = 18 \)
Wrapping ribbon: \( 2l + 2w + 4h = 2*12 + 2*10 + 4*18 = 24 + 20 + 72 = 116 \) cm
Bow: 30 cm
Total for Box 1: \( 116 + 30 = 146 \) cm

Step2: Calculate ribbon for Box 2

\( l = 30 \), \( w = 10 \), \( h = 10 \)
Wrapping ribbon: \( 2*30 + 2*10 + 4*10 = 60 + 20 + 40 = 120 \) cm
Bow: 30 cm
Total for Box 2: \( 120 + 30 = 150 \) cm

Step3: Calculate ribbon for Box 3

\( l = 25 \), \( w = 11 \), \( h = 16 \)
Wrapping ribbon: \( 2*25 + 2*11 + 4*16 = 50 + 22 + 64 = 136 \) cm
Bow: 30 cm
Total for Box 3: \( 136 + 30 = 166 \) cm

Step4: Total ribbon needed

\( 146 + 150 + 166 = 462 \) cm \( = 4.62 \) m

Step5: Compare with available ribbon (1.5 m = 150 cm)

\( 462 > 150 \), so not enough. But maybe the question is to find how much more ribbon is needed: \( 462 - 150 = 312 \) cm \( = 3.12 \) m

But since the problem is not fully stated, but based on the given, let's assume we need to calculate the total ribbon needed for the three boxes.

Answer:

The total ribbon needed for the three boxes is 462 centimeters (or 4.62 meters), and the available ribbon is 150 centimeters, so we need 312 more centimeters (or 3.12 more meters) of ribbon. (If the question was different, adjust accordingly, but based on the calculation, the total needed is 462 cm)