Math / Geometry

Question```latex The figure $ABCDEF$ , not drawn to scale, represents a wedge (prism) with measurements as shown. $BC$ is perpendicular to the plane $FEDC$ .
Given: - $BC = 5 \, \text{cm}$ - $DC = 12 \, \text{cm}$ - $ED = 13 \, \text{cm}$
Calculate: (i) the length, in cm, of $BD$ (ii) the surface area, in $\mathrm{cm}^{2}$ , of the wedge (iii) the volume of the prism ```

Studdy Solution

STEP 1

1. The wedge is a right triangular prism.
2. $BC$ is perpendicular to the plane $FEDC$ .
3. The base $FEDC$ is a rectangle with $ED = 13 \, \text{cm}$ and $DC = 12 \, \text{cm}$ .
4. $BC = 5 \, \text{cm}$ .

STEP 2

1. Calculate the length of $BD$ .
2. Calculate the surface area of the wedge.
3. Calculate the volume of the prism.

STEP 3

To find the length of $BD$ , use the Pythagorean theorem in triangle $BCD$ since $BC$ is perpendicular to $DC$ :
$BD^2 = BC^2 + DC^2$
Substitute the given values:
$BD^2 = 5^2 + 12^2$
Calculate:
$BD^2 = 25 + 144 = 169$
$BD = \sqrt{169} = 13 \, \text{cm}$

STEP 4

Calculate the surface area of the wedge. The surface area includes the areas of the rectangle $FEDC$ , two triangles $BCD$ and $AEF$ , and the rectangles $ABFE$ and $BCED$ .
1. Area of rectangle $FEDC$ :
$\text{Area}_{FEDC} = ED \times DC = 13 \, \text{cm} \times 12 \, \text{cm} = 156 \, \text{cm}^2$
2. Area of triangle $BCD$ :
$\text{Area}_{BCD} = \frac{1}{2} \times BC \times DC = \frac{1}{2} \times 5 \, \text{cm} \times 12 \, \text{cm} = 30 \, \text{cm}^2$
3. Area of rectangle $BCED$ :
$\text{Area}_{BCED} = BC \times ED = 5 \, \text{cm} \times 13 \, \text{cm} = 65 \, \text{cm}^2$
4. Area of rectangle $ABFE$ is the same as $BCED$ since it is parallel and congruent:
$\text{Area}_{ABFE} = 65 \, \text{cm}^2$
5. Area of triangle $AEF$ is the same as $BCD$ since it is congruent:
$\text{Area}_{AEF} = 30 \, \text{cm}^2$
Total surface area:
$\text{Surface Area} = \text{Area}_{FEDC} + 2 \times \text{Area}_{BCD} + \text{Area}_{BCED} + \text{Area}_{ABFE}$
$= 156 + 2 \times 30 + 65 + 65 = 346 \, \text{cm}^2$

STEP 5

Calculate the volume of the prism. The volume is the area of the base $FEDC$ times the height $BC$ :
$V = \text{Base Area} \times \text{Height}$
$= 156 \, \text{cm}^2 \times 5 \, \text{cm}$
$= 780 \, \text{cm}^3$
The solutions are: (i) The length of $BD$ is $\boxed{13 \, \text{cm}}$ . (ii) The surface area of the wedge is $\boxed{346 \, \text{cm}^2}$ . (iii) The volume of the prism is $\boxed{780 \, \text{cm}^3}$ .

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