Question

A cylindrical tube \(9.5 \mathrm{~cm}\) high and \(0.85 \mathrm{~cm}\) in diameter is used to collect blood samples. How many cubic decimeters (dm \(^{3}\) ) of blood can it hold ( \(V\) of a cylinder \(=\pi r^{2} h\) )?

Short answer

0.0036 cubic decimeters

Step-by-step solution

- Convert Diameter to Radius

The diameter of the cylindrical tube is 0.85 cm. The radius (r) is half of the diameter. Therefore, calculate the radius as follows:\(r = \frac{0.85 \, \text{cm}}{2} = 0.425 \, \text{cm}\)

- Calculate the Volume of the Cylinder in Cubic Centimeters

The formula to calculate the volume \(V\) of a cylinder is given by \(V = \pi r^{2} h\). Substitute the values of r and h:\(V = \pi (0.425 \, \text{cm})^{2} (9.5 \, \text{cm})\)First, calculate \(r^2\):\(0.425^{2} = 0.180625 \, \text{cm}^{2}\)Then calculate the entire volume:\(V = \pi (0.180625) (9.5) \approx 3.6 \, \text{cm}^{3}\)

- Convert Cubic Centimeters to Cubic Decimeters

Since there are 1000 cubic centimeters in a cubic decimeter, convert the volume from cubic centimeters to cubic decimeters:\(V = \frac{3.6 \, \text{cm}^{3}}{1000} = 0.0036 \, \text{dm}^{3}\)

Key concepts

Cylindrical Volume Formula

To calculate the volume of a cylinder, we use the formula: \( V = \pi r^{2} h \). This formula tells us that the volume is determined by multiplying the area of the base (which is a circle) by the height of the cylinder.

Here's a simple breakdown:

• \( \pi \footnote\) is a mathematical constant, approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter.
• \( r \footnote\) is the radius of the cylinder's base.
• \( r^2 \footnote\) is the radius squared, which gives the area of the base circle.
• \( h \footnote\) is the height of the cylinder.

By substituting these values into our formula, we can easily determine the volume of any cylindrical object.

This approach is crucial in many practical applications, from calculating how much liquid a container can hold to determining material usage for manufacturing. Knowing this formula helps solve many real-world problems involving cylindrical shapes.

Unit Conversion

Understanding how to convert between different units is essential for solving many math and science problems. In this exercise, we converted cubic centimeters (cm³) to cubic decimeters (dm³).

Why is this important? Because units must be consistent in calculations.

Steps for conversion in our example:

• The cylinder's volume was first calculated in cubic centimeters.
• Since 1 cubic decimeter (dm³) is equivalent to 1000 cubic centimeters (cm³), we divide the volume in cubic centimeters by 1000 to get the volume in cubic decimeters.

So, our final volume in cubic decimeters is \( \frac{3.6 \text{cm}^3}{1000} = 0.0036 \text{dm}^3 \footnote\).

By mastering unit conversion, students can handle diverse problems more effectively, ensuring that the units they work with are consistent across different contexts.

Radius and Diameter in Geometry

In geometry, understanding the difference between radius and diameter is fundamental.

Here's a quick rundown:

• The diameter (\footnote\text{d\footnote\}) of a circle is the distance across the circle, passing through its center.
• The radius (\footnote\text{r\footnote\}) is half the diameter, extending from the center of the circle to any point on its perimeter.
• Mathematically, this relationship is expressed as r = \(\frac{\text{d}}{2} \footnote\).

In our exercise, we convert the diameter of the cylinder's base to its radius:

The given diameter is 0.85 cm, so the radius is \(r = \frac{0.85 \text{cm}}{2} = 0.425 \text{cm} \footnote\).

Knowing how to switch between these two measurements helps in applying the cylindrical volume formula accurately.

Grasping these fundamental concepts in geometry not only simplifies volume calculations but also aids in understanding the properties and relationships of different shapes.