360 cubes of 1cm are arranged to make a cuboid. The largest base area of the cuboid is a perfect square. Find the smallest possible height of the cuboid. Given 4k + 3r2 = 10, make r2 the subject of the formula.

here are the solutions:

Question (c)

Step 1: Determine the total volume of the cuboid. Since 360 cubes of 1 cm are used, the volume of each cube is $1 cm \times 1 cm \times 1 cm = 1 cm^3$. The total volume of the cuboid is $360 \times 1 cm^3 = 360 cm^3$.

Step 2: Understand the relationship between volume, base area, and height. The volume of a cuboid is given by $V = Base Area \times \text{Height}$. So, $\text{Height} = \frac{Volume}{Base Area}$. To find the smallest possible height, we need to find the largest possible base area.

Step 3: Identify the largest base area that is a perfect square and a factor of the volume. The volume is $360 cm^3$. We need to find the largest perfect square that divides 360. Let's list the perfect square factors of 360:

• $1^2 = 1$ (360 is divisible by 1)
• $2^2 = 4$ (360 is divisible by 4, $360 \div 4 = 90$)
• $3^2 = 9$ (360 is divisible by 9, $360 \div 9 = 40$)
• $4^2 = 16$ (360 is not divisible by 16)
• $5^2 = 25$ (360 is not divisible by 25)
• $6^2 = 36$ (360 is divisible by 36, $360 \div 36 = 10$) The next perfect square is $7^2 = 49$, which is not a factor of 360. The largest perfect square factor of 360 is 36. Therefore, the largest base area is $36 cm^2$.

Step 4: Calculate the smallest possible height. $\text{Height} = \frac{Volume}{Base Area} = \frac{360 cm^3}{36 cm^2} = 10 \text{ cm}$

The smallest possible height of the cuboid is $\boxed{10 cm}$.

Question (d)

Step 1: Start with the given equation. $4k + 3r^2 = 10$

Step 2: Isolate the term containing $r^2$ by subtracting $4k$ from both sides. $3r^2 = 10 - 4k$

Step 3: Make $r^2$ the subject by dividing both sides by 3. $r^2 = \frac{10 - 4k}{3}$

The formula with $r^2$ as the subject is $\boxed{r^2 = \frac{10 - 4k}{3}}$.