A geodesic dome constructed with an aluminum framework is a nearly perfect hemisphere; its diameter measures 55.0 m on a winter day at a temperature of -15°C. How much more interior space does the dome have in the summer, when the temperature is 35°C?

Video transcript

Welcome back everybody. We are taking a look at a shot from a shot put event. Now a shot is just a spherical mass that is to be thrown and we are told that the diameter of this shot is 130. We're told that it is this diane At a temperature of 20°C. Now we are tasked with finding what the change in volume is of the shot when stored in a summer location, a more summery location where the temperature is 50°C. So we have a formula for the change of volume. This is going to be equal to the coefficient of volume expansion times our initial volume times our change in temperature. Now we need to know some of these values before we can plug this into the formula, I can give you this our beta or coefficient of volume expansion for brass, which is what our shot is made out of is six times 10 to the fifth Celsius to the negative one. What about our change in temperature or a change in temperature is just equal to our final temperature minus our initial temperature. So this is going to be 50 minus 20 which equals 30. So now we have these two terms but we still need to find what our original volume was. Well, we know that this shot is spherical and the formula for the volume of a sphere is four thirds pi r cubed. So let's go ahead and plug in our our diameter into this but then make sure we convert it to a radius and simply we do that by dividing our diameter by two. So 1 30 divided by two will be our radius. We cube this and when you plug this into your calculator, you get that. Our initial volume was 7.963 times 10 to the fifth. Now we have everything to find the change in volume of the shot. So let's go ahead and plug everything in. We have Our coefficient of volume expansion to be six times 10 to the negative, 5th times 7.963 times 10 to the fifth which was our volume times are changing temperature of 30 degrees. And when you plug all of this into your calculator, you get that our change in volume of the shot, 1433 millimeters cubed, which corresponds to answer choice B. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.