To solve the problem of finding the lower bound for the height of the cuboid, let's work through the problem step-by-step.
1. Understand the given values:
- Volume of the cuboid: [tex]\( 878 \, \text{cm}^3 \)[/tex] (correct to the nearest cubic centimetre)
- Length of the base: [tex]\( 7 \, \text{cm} \)[/tex] (correct to the nearest centimetre)
- Width of the base: [tex]\( 6 \, \text{cm} \)[/tex] (correct to the nearest centimetre)
2. Determine the bounds for the given dimensions of the cuboid:
- For a measurement given correct to the nearest unit, the actual value can be anywhere within 0.5 units below to 0.5 units above the stated value.
Length bounds:
- Lower bound: [tex]\( 7 \, \text{cm} - 0.5 \, \text{cm} = 6.5 \, \text{cm} \)[/tex]
- Upper bound: [tex]\( 7 \, \text{cm} + 0.5 \, \text{cm} = 7.5 \, \text{cm} \)[/tex]
Width bounds:
- Lower bound: [tex]\( 6 \, \text{cm} - 0.5 \, \text{cm} = 5.5 \, \text{cm} \)[/tex]
- Upper bound: [tex]\( 6 \, \text{cm} + 0.5 \, \text{cm} = 6.5 \, \text{cm} \)[/tex]
3. Calculate the upper and lower bounds for the cross-sectional area of the base:
- The largest possible value of the area is given by the upper bounds of length and width.
- So, the area of the base using the upper bounds is [tex]\( 7.5 \, \text{cm} \times 6.5 \, \text{cm} \)[/tex].
4. Volume calculation to find the height:
- The height of the cuboid can be calculated using the formula: [tex]\( \text{Height} = \frac{\text{Volume}}{\text{Area of base}} \)[/tex].
- To find the lower bound for the height, use the smallest possible values (i.e., the upper bounds for length and width) to get the largest area of the base.
- Thus, the lower bound for the height is calculated as follows:
[tex]\[
\text{Height}_{\text{lower bound}} = \frac{878 \, \text{cm}^3}{7.5 \, \text{cm} \times 6.5 \, \text{cm}}
\][/tex]
Plugging in the numbers:
[tex]\[
7.5 \times 6.5 = 48.75 \, \text{cm}^2
\][/tex]
[tex]\[
\text{Height}_{\text{lower bound}} = \frac{878 \, \text{cm}^3}{48.75 \, \text{cm}^2} \approx 18.01025641025641 \, \text{cm}
\][/tex]
Therefore, the lower bound for the height of the cuboid is approximately [tex]\( 18.01025641025641 \, \text{cm} \)[/tex].
