Nicole C. answered • 09/18/15
Math, Science, Engineering and Test Prep
For this problem we want to first gain an understand of what the shape looks like we can properly dissect it. a cone with an attached hemisphere will look like the images in the following link: https://www.google.com/search?q=cone+attached+hemisphere&espv=2&biw=1249&bih=638&source=lnms&tbm=isch&sa=X&ved=0CAYQ_AUoAWoVChMI0LCb3-P_xwIV1RaSCh14UQC1 From that we can see the total shape can be broken up into a cone and half a sphere. Thus to find the volumes we need to know the formula for the volume of a cone and of a sphere. V_cone = pi*r^2*h/3 V_sphere = (4/3)*pi*r^3 V_hemisphere = (2/3)*pi*r^3 For part a, we simply plug and chug the respective height and radius into the formulas to solve V_cone + V_hemisphere = pi*r^2*h/3 + (2/3)*pi*r^3 V_cone + V_hemisphere = pi*(2.5)^2*(21)/3 + (2/3)*pi*(2.5)^3 I leave it to you to simply the expression. For part b we will want to replace r with 2r and h with 2h where ever it appears. 2V_cone + 2V_hemishpere = pi*(2r)^2*(2h)/3 + (2/3)*pi*(2r)^3 2V_cone + 2V_hemishpere = pi*2^2*r^2*2h/3 + (2/3)*pi*2^3*r^3 2V_cone + 2V_hemishpere = pi*4r^2*2h/3 + (2/3)*pi*8r^3 Now ask yourself if there is a number that can be factored out of all the expressions such that: 2V_cone + 2V_hemishpere = __ * [pi*r^2*h/3 + (2/3)*pi*r^3] That number is how much the volume is scaled up when length is scaled up by 2. We will follow a similar process for part c. (1/4)V_cone + (1/4)V_hemishpere = pi*(r/4)^2*(h/4)/3 + (2/3)*pi*(r/4)^3 (1/4)V_cone + (1/4)V_hemishpere = pi*r^2/4^2*(h/4)/3 + (2/3)*pi**r^3/4^3 (1/4)V_cone + (1/4)V_hemishpere = pi*r^2/16*(h/4)/3 + (2/3)*pi**r^3/64 Again what number can be factored out such that: (1/4)V_cone + (1/4)V_hemishpere = __ * [pi*(r/4)^2*(h/4)/3 + (2/3)*pi*(r/4)^3] Hint: this time you will be looking for a fraction that will be the scale down factor.
