A solid cylindrical metallic rod 15-cm long with a diameter equal to 3.5 cm is melted intoanother rod of radius 3.5 cm. What will be the length of the new rod?

Sure, let's solve this step by step.

Step 1: First, we need to find the volume of the original cylindrical rod. The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height (or in this case, the length of the rod). The radius is half the diameter, so r = 3.5 cm / 2 = 1.75 cm. The length of the rod is 15 cm. So, the volume of the original rod is V = π(1.75 cm)²(15 cm).

Step 2: The problem states that the original rod is melted down and reshaped into a new rod with a radius of 3.5 cm. The volume of the new rod will be the same as the volume of the original rod, because no material is lost in the process.

Step 3: We can find the length of the new rod by rearranging the formula for the volume of a cylinder to solve for h: h = V / (πr²). We know that V is the same for both rods, and r for the new rod is 3.5 cm. So, the length of the new rod is h = V / (π(3.5 cm)²).

By calculating the above steps, we can find the length of the new rod.

Sure, let's solve this step by step.

Step 1: Calculate the volume of the original rod. The volume V of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height (or in this case, the length of the rod). The radius is half the diameter, so for the original rod, r = 3.5 cm / 2 = 1.75 cm. The length is 15 cm. So, the volume of the original rod is V = π(1.75 cm)²(15 cm).

Step 2: Set up the equation for the volume of the new rod. The new rod has a radius of 3.5 cm and an unknown length, which we'll call L. So, its volume is V = π(3.5 cm)²L.

Step 3: Since the volume of the metal doesn't change when the rod is melted and reshaped, we can set the two volume equations equal to each other and solve for L: π(1.75 cm)²(15 cm) = π(3.5 cm)²L

Step 4: Solve for L. First, we can cancel out the π on both sides. Then, we can divide both sides by (3.5 cm)² to solve for L: L = (1.75 cm / 3.5 cm)² * 15 cm

Step 5: Calculate the length L. L = (0.5)² * 15 cm = 0.25 * 15 cm = 3.75 cm

So, the length of the new rod will be 3.75 cm.