If the diameter of a youth baseball is 2.8 inches and the diameter of an adult softball is 3.8 inches, what is the approximate difference in their volumes? Use 3.14 for [tex]$\pi$[/tex]. Round to the nearest tenth.
Recall the formula for the volume of a sphere: [tex]$V=\frac{4}{3} \pi r^3$[/tex].
A. 1.0 cubic inch
B. 17.2 cubic inches
C. 40.2 cubic inches
D. 137.8 cubic inches
Answer (2)
Calculate the radius of the youth baseball: r 1 = 2 2.8 = 1.4 inches.
Calculate the radius of the adult softball: r 2 = 2 3.8 = 1.9 inches.
Calculate the volumes of both the baseball and the softball: V 1 = 11.48821333 cubic inches and V 2 = 28.71634667 cubic inches.
Find the difference in their volumes and round to the nearest tenth: 17.2 cubic inches.
Explanation
Problem Analysis First, let's identify the information we have. We know the diameter of a youth baseball is 2.8 inches and the diameter of an adult softball is 3.8 inches. We are asked to find the approximate difference in their volumes, using π = 3.14 , and rounding the final answer to the nearest tenth. We also recall the formula for the volume of a sphere: V = 3 4 π r 3 .
Calculate Radii Next, we need to find the radii of both the baseball and the softball. The radius is half of the diameter. So:
Radius of youth baseball, r 1 = 2 2.8 = 1.4 inches Radius of adult softball, r 2 = 2 3.8 = 1.9 inches
Calculate Volumes Now we can calculate the volumes of both the baseball and the softball using the formula V = 3 4 π r 3 :
Volume of youth baseball, V 1 = 3 4 × 3.14 × ( 1.4 ) 3 Volume of adult softball, V 2 = 3 4 × 3.14 × ( 1.9 ) 3
Compute Volumes Let's calculate V 1 and V 2 :
V 1 = 3 4 × 3.14 × ( 1.4 ) 3 = 3 4 × 3.14 × 2.744 = 11.48821333... cubic inches V 2 = 3 4 × 3.14 × ( 1.9 ) 3 = 3 4 × 3.14 × 6.859 = 28.71634667... cubic inches
Find the Difference Now, we find the difference in their volumes:
V 2 − V 1 = 28.71634667 − 11.48821333 = 17.22813333 cubic inches
Round the Answer Finally, we round the difference to the nearest tenth:
17.22813333 ≈ 17.2 cubic inches
Final Answer Therefore, the approximate difference in their volumes is 17.2 cubic inches.
Examples
Understanding the volume of spheres is useful in many real-world applications. For example, when designing sports equipment, knowing the volume helps determine the amount of material needed and affects the object's weight and performance. In packaging, calculating the volume of spherical items ensures appropriate box sizes, reducing waste and shipping costs. Even in cooking, understanding volume helps in measuring ingredients accurately, especially when dealing with spherical fruits or vegetables.
The approximate difference in the volumes of a youth baseball and an adult softball is 17.2 cubic inches. This was calculated using the formula for the volume of a sphere, with given diameters converted to radii. The calculations showed that the youth baseball's volume is about 11.5 cubic inches and the softball's volume is about 28.7 cubic inches.
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