Answer :
To solve these problems, we need to approach them step-by-step:
4. How many cubes of side 15 cm can be fitted into a box which measures 1.5m × 90cm × 75cm?
First, let's convert all dimensions to centimeters, as the cube's side length is given in centimeters. Note that 1 meter is 100 centimeters.
- The box's length is 1.5 meters, which is 150 cm.
- The box's width is 90 cm.
- The box's height is 75 cm.
The volume of one cube with side 15 cm is:
[tex]\text{Volume of cube} = (15 \text{ cm})^3 = 3375 \text{ cm}^3[/tex]
The volume of the box is:
[tex]\text{Volume of box} = 150 \text{ cm} \times 90 \text{ cm} \times 75 \text{ cm} = 1012500 \text{ cm}^3[/tex]
Now, we calculate how many such cubes can be placed inside this box by dividing the volume of the box by the volume of one cube:
[tex]\text{Number of cubes} = \frac{1012500 \text{ cm}^3}{3375 \text{ cm}^3} = 300[/tex]
Thus, we can fit 300 cubes of side 15 cm into the box.
5. Find the value of [tex]x[/tex] for which [tex]2x + 2^{-4} = 2^5[/tex].
Let's simplify and solve the equation for [tex]x[/tex]:
The given equation is:
[tex]2x + 2^{-4} = 2^5[/tex]
First, let's evaluate [tex]2^{-4}[/tex]:
[tex]2^{-4} = \frac{1}{2^4} = \frac{1}{16}[/tex]
Substituting this back into the equation gives:
[tex]2x + \frac{1}{16} = 32[/tex]
We know [tex]2^5 = 32[/tex], so the equation becomes:
[tex]2x = 32 - \frac{1}{16}[/tex]
Converting 32 to a fraction over 16 to subtract easily:
[tex]32 = \frac{512}{16}[/tex]
Thus:
[tex]2x = \frac{512}{16} - \frac{1}{16} = \frac{511}{16}[/tex]
Divide both sides by 2 to solve for [tex]x[/tex]:
[tex]x = \frac{511}{16} \times \frac{1}{2} = \frac{511}{32}[/tex]
Hence, the value of [tex]x[/tex] is [tex]\frac{511}{32}[/tex].
