Q15. Consider a box with dimensions 3 cm × 5 cm × 11 cm. If all of its dimensions are increased by x cm, what values of x will give a box with a volume between 300 cm3 and 900 cm3?
To find the volume of the box before increasing its dimensions, we multiply its length, width, and height:
Volume before = 3 cm × 5 cm × 11 cm = 165 cm³
Now, let's find the volume after increasing its dimensions by x cm:
Length after = 3 cm + x cm
Width after = 5 cm + x cm
Height after = 11 cm + x cm
Volume after = (3 cm + x cm) × (5 cm + x cm) × (11 cm + x cm)
Since we want the volume to be between 300 cm³ and 900 cm³:
300 cm³ < (3 cm + x cm) × (5 cm + x cm) × (11 cm + x cm) < 900 cm³
Now, we can solve this inequality:
300 cm³ < (3 cm + x cm) × (5 cm + x cm) × (11 cm + x cm) < 900 cm³
Expanding the factors:
300 cm³ < (15 cm² + 8 cmx + x²) × (11 cm + x cm) < 900 cm³
Multiplying the binomials:
300 cm³ < 165 cm² + 91 cmx + 11 x² + 8 cmx + 8x² + x³ < 900 cm³
Combining like terms:
300 cm³ < 165 cm² + 99 cmx + 19 x² + x³ < 900 cm³
Now, we can factor and solve the inequality by considering each part:
The volume must be greater than 300 cm³:
165 cm² + 99 cmx + 19 x² + x³ - 300 cm³ > 0
The volume must be less than 900 cm³:
165 cm² + 99 cmx + 19 x² + x³ - 900 cm³ < 0
To find the values of x, we can use a graphing calculator or factoring techniques. The solutions for both parts of the inequality will give us the range of x values that will give a box with a volume between 300 cm³ and 900 cm³.
To find the possible values of x that will give a box with a volume between 300 cm³ and 900 cm³, we can use the formula for the volume of a rectangular box:
Volume = Length x Width x Height
Given that the original dimensions of the box are 3 cm x 5 cm x 11 cm, we can calculate the original volume:
Original Volume = 3 cm x 5 cm x 11 cm = 165 cm³
Now, let's increase all the dimensions by x cm:
New Length = 3 cm + x cm
New Width = 5 cm + x cm
New Height = 11 cm + x cm
The new volume can be calculated as:
New Volume = New Length x New Width x New Height
Substituting in the new dimensions, we get:
New Volume = (3 cm + x cm) * (5 cm + x cm) * (11 cm + x cm)
To find the possible values of x, we need to solve the inequality:
300 cm³ < New Volume < 900 cm³
Let's first solve for the lower bound:
300 cm³ < (3 cm + x) * (5 cm + x) * (11 cm + x)
300 cm³ < (15 cm + 3x) * (11 cm + x)
300 cm³ < 165 cm² + 18 cmx + 33 cx + 3x² + 5x² + x³
0 < x³ + 8x² + 51 cmx - 135 cm²
Now, let's solve for the upper bound:
(3 cm + x) * (5 cm + x) * (11 cm + x) < 900 cm³
165 cm² + 18 cmx + 33 cmx + 3x² + 5x² + x³ < 900 cm³
We now have a cubic inequality, and we can solve it to find the values of x that satisfy the conditions. However, solving cubic inequalities can be complex and involve finding the roots of the cubic equation.
Therefore, it is recommended to use numerical methods or calculators to find the exact values of x that satisfy the conditions.