Answer :
To find the height of the cuboid, we can follow a step-by-step procedure using the given information:
1. Given Information:
- Volume (V) of the cuboid = 300 cm³
- Surface area (SA) of the cuboid = 370 cm²
- Length (L) of the cuboid = 20 cm
2. Formulating Equations:
Let the width of the cuboid be [tex]\( W \)[/tex] and the height be [tex]\( H \)[/tex].
- The volume of the cuboid is given by:
[tex]\[ V = L \times W \times H \][/tex]
Substituting the given values, we have:
[tex]\[ 300 = 20 \times W \times H \][/tex]
- Solving for [tex]\( W \times H \)[/tex]:
[tex]\[ W \times H = \frac{300}{20} = 15 \quad (1) \][/tex]
- The surface area of the cuboid is given by:
[tex]\[ SA = 2(L \times W + W \times H + H \times L) \][/tex]
Substituting the given values, we have:
[tex]\[ 370 = 2(20 \times W + W \times H + H \times 20) \][/tex]
- Simplified, this becomes:
[tex]\[ 370 = 40W + 2WH + 40H \][/tex]
Dividing the entire equation by 2, we get:
[tex]\[ 185 = 20W + WH + 20H \quad (2) \][/tex]
3. Solving the Equations:
From equation (1), we have [tex]\( W \times H = 15 \)[/tex]. Solving for [tex]\( W \)[/tex]:
[tex]\[ W = \frac{15}{H} \][/tex]
Substituting [tex]\( W \)[/tex] in equation (2):
[tex]\[ 185 = 20\left(\frac{15}{H}\right) + \frac{15}{H} \times H + 20H \][/tex]
Simplifying, we get:
[tex]\[ 185 = 300/H + 15 + 20H \][/tex]
Rearranging the equation gives us:
[tex]\[ 185 - 15 = \frac{300}{H} + 20H \][/tex]
[tex]\[ 170 = \frac{300}{H} + 20H \][/tex]
4. Multiplying through by [tex]\( H \)[/tex] to eliminate the fraction:
[tex]\[ 170H = 300 + 20H^2 \][/tex]
Rearranging into a standard quadratic equation:
[tex]\[ 20H^2 + 300 - 170H = 0 \][/tex]
[tex]\[ 20H^2 - 170H + 300 = 0 \][/tex]
5. Solving the Quadratic Equation:
We can use the quadratic formula [tex]\( H = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 20 \)[/tex], [tex]\( b = -170 \)[/tex], and [tex]\( c = 300 \)[/tex].
[tex]\[ H = \frac{-(-170) \pm \sqrt{(-170)^2 - 4 \times 20 \times 300}}{2 \times 20} \][/tex]
[tex]\[ H = \frac{170 \pm \sqrt{28900 - 24000}}{40} \][/tex]
[tex]\[ H = \frac{170 \pm \sqrt{4900}}{40} \][/tex]
[tex]\[ H = \frac{170 \pm 70}{40} \][/tex]
This yields two potential solutions:
[tex]\[ H_1 = \frac{240}{40} = 6 \quad \text{and} \quad H_2 = \frac{100}{40} = 2.5 \][/tex]
6. Identifying the Correct Height:
We know that the width [tex]\( W \)[/tex] must be greater than the height [tex]\( H \)[/tex]. Checking both solutions:
- If [tex]\( H = 6 \)[/tex]:
[tex]\[ W = \frac{15}{H} = \frac{15}{6} = 2.5 \quad \text{(not valid as W < H)} \][/tex]
- If [tex]\( H = 2.5 \)[/tex]:
[tex]\[ W = \frac{15}{H} = \frac{15}{2.5} = 6 \quad \text{(valid as W > H)} \][/tex]
Thus, the height [tex]\( H \)[/tex] of the cuboid is:
[tex]\[ H = 2.5 \, \text{cm} \][/tex]
1. Given Information:
- Volume (V) of the cuboid = 300 cm³
- Surface area (SA) of the cuboid = 370 cm²
- Length (L) of the cuboid = 20 cm
2. Formulating Equations:
Let the width of the cuboid be [tex]\( W \)[/tex] and the height be [tex]\( H \)[/tex].
- The volume of the cuboid is given by:
[tex]\[ V = L \times W \times H \][/tex]
Substituting the given values, we have:
[tex]\[ 300 = 20 \times W \times H \][/tex]
- Solving for [tex]\( W \times H \)[/tex]:
[tex]\[ W \times H = \frac{300}{20} = 15 \quad (1) \][/tex]
- The surface area of the cuboid is given by:
[tex]\[ SA = 2(L \times W + W \times H + H \times L) \][/tex]
Substituting the given values, we have:
[tex]\[ 370 = 2(20 \times W + W \times H + H \times 20) \][/tex]
- Simplified, this becomes:
[tex]\[ 370 = 40W + 2WH + 40H \][/tex]
Dividing the entire equation by 2, we get:
[tex]\[ 185 = 20W + WH + 20H \quad (2) \][/tex]
3. Solving the Equations:
From equation (1), we have [tex]\( W \times H = 15 \)[/tex]. Solving for [tex]\( W \)[/tex]:
[tex]\[ W = \frac{15}{H} \][/tex]
Substituting [tex]\( W \)[/tex] in equation (2):
[tex]\[ 185 = 20\left(\frac{15}{H}\right) + \frac{15}{H} \times H + 20H \][/tex]
Simplifying, we get:
[tex]\[ 185 = 300/H + 15 + 20H \][/tex]
Rearranging the equation gives us:
[tex]\[ 185 - 15 = \frac{300}{H} + 20H \][/tex]
[tex]\[ 170 = \frac{300}{H} + 20H \][/tex]
4. Multiplying through by [tex]\( H \)[/tex] to eliminate the fraction:
[tex]\[ 170H = 300 + 20H^2 \][/tex]
Rearranging into a standard quadratic equation:
[tex]\[ 20H^2 + 300 - 170H = 0 \][/tex]
[tex]\[ 20H^2 - 170H + 300 = 0 \][/tex]
5. Solving the Quadratic Equation:
We can use the quadratic formula [tex]\( H = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 20 \)[/tex], [tex]\( b = -170 \)[/tex], and [tex]\( c = 300 \)[/tex].
[tex]\[ H = \frac{-(-170) \pm \sqrt{(-170)^2 - 4 \times 20 \times 300}}{2 \times 20} \][/tex]
[tex]\[ H = \frac{170 \pm \sqrt{28900 - 24000}}{40} \][/tex]
[tex]\[ H = \frac{170 \pm \sqrt{4900}}{40} \][/tex]
[tex]\[ H = \frac{170 \pm 70}{40} \][/tex]
This yields two potential solutions:
[tex]\[ H_1 = \frac{240}{40} = 6 \quad \text{and} \quad H_2 = \frac{100}{40} = 2.5 \][/tex]
6. Identifying the Correct Height:
We know that the width [tex]\( W \)[/tex] must be greater than the height [tex]\( H \)[/tex]. Checking both solutions:
- If [tex]\( H = 6 \)[/tex]:
[tex]\[ W = \frac{15}{H} = \frac{15}{6} = 2.5 \quad \text{(not valid as W < H)} \][/tex]
- If [tex]\( H = 2.5 \)[/tex]:
[tex]\[ W = \frac{15}{H} = \frac{15}{2.5} = 6 \quad \text{(valid as W > H)} \][/tex]
Thus, the height [tex]\( H \)[/tex] of the cuboid is:
[tex]\[ H = 2.5 \, \text{cm} \][/tex]
