Question

1. calculer le volume dexcavation dune maison de 38 x 28 avec une profondeur dexcavation de 8-6\ avec une pente 1 dans 1 sachant que : la projection de la semelle est de 8\ et on tiendra compte de lespace du drain français autour du bâtiment est de 16\ :

Explanation:

Step1: Convert all units to inches

First, convert feet to inches (1 foot = 12 inches):

• Building dimensions: $38' = 38\times12 = 456''$, $28' = 28\times12 = 336''$
• Excavation depth: $8'-6'' = (8\times12)+6 = 102''$
• Footing projection: $8''$, French drain space: $16''$

Step2: Calculate total side extensions

Total extension per side = footing projection + drain space = $8'' + 16'' = 24''$ (2 feet)
Total extended length: $456'' + 2\times24'' = 504'' = 42'$
Total extended width: $336'' + 2\times24'' = 384'' = 32'$

Step3: Calculate top excavation area

Top area of excavation: $A_{top} = 42' \times 32' = 1344\ \text{sq ft}$

Step4: Calculate bottom excavation dimensions

Since slope is 1:1, the width/length decreases by 2×depth (1ft inward per 1ft depth):
Bottom length: $42' - 2\times8.5' = 42' - 17' = 25'$
Bottom width: $32' - 2\times8.5' = 32' - 17' = 15'$
Bottom area: $A_{bottom} = 25' \times 15' = 375\ \text{sq ft}$

Step5: Calculate average area and volume

Use frustum volume formula $V = \frac{h}{3}(A_{top} + A_{bottom} + \sqrt{A_{top}A_{bottom}})$
First, compute $\sqrt{A_{top}A_{bottom}} = \sqrt{1344\times375} = \sqrt{504000} \approx 710$
Average area component: $1344 + 375 + 710 = 2429$
Volume: $V = \frac{8.5}{3} \times 2429 \approx 8.5 \times 809.67 \approx 6882.2\ \text{cu ft}$

Answer:

The excavation volume is approximately $\boldsymbol{6882\ \text{cubic feet}}$ (or ~254.9 cubic yards, if converted: $\frac{6882}{27} \approx 254.9$)