If the diameter of a youth softball is 3.5 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 of a cubic inch.
Recall the formula Sphere [tex]$V=\frac{4}{3} \pi r^3$[/tex].
A. 6.3 cubic inches
B. 51.1 cubic inches
C. 67.0 cubic inches
D. 409.2 cubic inches
Answer (2)
Calculate the radii of the youth and adult softballs by dividing their diameters by 2: r y = 1.75 inches and r a = 1.9 inches.
Compute the volumes of both softballs using the sphere volume formula V = 3 4 π r 3 , resulting in V y = 22.4379 cubic inches and V a = 28.7163 cubic inches.
Determine the difference in volumes by subtracting the youth softball's volume from the adult softball's volume: V a − V y = 6.2784 cubic inches.
Round the volume difference to the nearest tenth of a cubic inch, yielding the final answer: 6.3 cubic inches.
Explanation
Identify Given Information First, let's identify the information we have:
Diameter of youth softball = 3.5 inches
Diameter of adult softball = 3.8 inches
π = 3.14
Volume of a sphere: V = 3 4 π r 3 , where r is the radius.
Calculate Radii Next, we need to find the radii of both softballs. Remember that the radius is half of the diameter.
Radius of youth softball ( r y ) = 2 3.5 = 1.75 inches
Radius of adult softball ( r a ) = 2 3.8 = 1.9 inches
Calculate Volumes Now, we can calculate the volumes of both softballs using the formula V = 3 4 π r 3 .
Volume of youth softball ( V y ) = 3 4 × 3.14 × ( 1.75 ) 3
V y = 3 4 × 3.14 × 5.359375
V y = 22.437916666666666 cubic inches
Volume of adult softball ( V a ) = 3 4 × 3.14 × ( 1.9 ) 3
V a = 3 4 × 3.14 × 6.859
V a = 28.716346666666663 cubic inches
Find the Difference To find the difference in their volumes, we subtract the volume of the youth softball from the volume of the adult softball:
Difference = V a − V y = 28.716346666666663 − 22.437916666666666 = 6.278429999999997 cubic inches
Round the Answer Finally, we round the difference to the nearest tenth of a cubic inch:
6.278429999999997 ≈ 6.3 cubic inches
State the Final Answer Therefore, the approximate difference in their volumes is 6.3 cubic inches.
Examples
Understanding the volume of spheres is useful in many real-world applications. For example, when manufacturing balls for sports, knowing the precise volume helps in determining the amount of material needed. In the medical field, calculating the volume of spherical tumors can help doctors monitor their growth and determine the effectiveness of treatments. Also, in engineering, calculating the volume of spherical tanks is essential for storing liquids and gases efficiently. These calculations ensure accuracy and help optimize resource usage in various fields.
The difference in volumes between a youth softball and an adult softball is approximately 6.3 cubic inches. This is calculated using the sphere volume formula by determining the radii of both softballs and computing their respective volumes. The final answer choice is A. 6.3 cubic inches.
;
