High School

4. How many cubes of side 15 cm can be fitted into a box which measures 1.5m × 90cm × 75cm?

5. Find the value of x for which 2x + 2^{-4} = 2^5.

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].