Question
Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in.3 The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning in easy words without the / and ()
Answers
GPT 3.5
To solve this problem, we need to find the volume of the ice cream packed into the cone and the volume of the hemisphere on top of the cone, and then add these two volumes together.
Let's start by finding the volume of the cone. The formula for the volume of a cone is V = (1/3)πr²h, where V is the volume, π is Pi (approximately 3.14), r is the radius of the base of the cone, and h is the height of the cone.
In this case, the diameter of the cone is equal to its height, so we can call the radius of the base r and the height h. Because the radius is half of the diameter, we have r = h/2.
Now we can plug these values into the formula for the volume of the cone: V_cone = (1/3)π(h/2)²h.
Next, let's find the volume of the hemisphere. The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius of the sphere.
In this case, the hemisphere is a half of a sphere, so we can use half the volume formula: V_hemisphere = (1/2) * (4/3)π(h/2)³.
Simplifying this formula, we have V_hemisphere = (2/3)π(h/2)³.
Now let's add the volumes of the cone and the hemisphere together to find the total volume of ice cream: V_total = V_cone + V_hemisphere.
Substituting the formulas we found earlier, we have V_total = (1/3)π(h/2)²h + (2/3)π(h/2)³.
To make calculation easier, we can simplify the formulas by canceling out common factors. Since both terms contain (h/2)² and (h/2)³, we can factor out (h/2)².
Simplifying further, we have V_total = [(1/3)π(h/2)² + (2/3)π(h/2)²] * h.
Combining the two terms, we get V_total = [(1/3 + 2/3)π(h/2)²] * h.
Simplifying the fraction, we have V_total = (1π(h/2)²) * h.
Finally, substituting the given value for the volume of the hemisphere (4in³), we have (1π(h/2)²) * h = 4.
Simplifying this equation and solving for h, we can find the height of the cone. After finding the height, we can substitute it back into the equation for the total volume to find the final answer.
Let's start by finding the volume of the cone. The formula for the volume of a cone is V = (1/3)πr²h, where V is the volume, π is Pi (approximately 3.14), r is the radius of the base of the cone, and h is the height of the cone.
In this case, the diameter of the cone is equal to its height, so we can call the radius of the base r and the height h. Because the radius is half of the diameter, we have r = h/2.
Now we can plug these values into the formula for the volume of the cone: V_cone = (1/3)π(h/2)²h.
Next, let's find the volume of the hemisphere. The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius of the sphere.
In this case, the hemisphere is a half of a sphere, so we can use half the volume formula: V_hemisphere = (1/2) * (4/3)π(h/2)³.
Simplifying this formula, we have V_hemisphere = (2/3)π(h/2)³.
Now let's add the volumes of the cone and the hemisphere together to find the total volume of ice cream: V_total = V_cone + V_hemisphere.
Substituting the formulas we found earlier, we have V_total = (1/3)π(h/2)²h + (2/3)π(h/2)³.
To make calculation easier, we can simplify the formulas by canceling out common factors. Since both terms contain (h/2)² and (h/2)³, we can factor out (h/2)².
Simplifying further, we have V_total = [(1/3)π(h/2)² + (2/3)π(h/2)²] * h.
Combining the two terms, we get V_total = [(1/3 + 2/3)π(h/2)²] * h.
Simplifying the fraction, we have V_total = (1π(h/2)²) * h.
Finally, substituting the given value for the volume of the hemisphere (4in³), we have (1π(h/2)²) * h = 4.
Simplifying this equation and solving for h, we can find the height of the cone. After finding the height, we can substitute it back into the equation for the total volume to find the final answer.