I am sooo confused on my homework.
#1 Two cars lave an intersection at the same time. One is headed south at a constant speed of 30mph. the other is heading west at a constant speed of 40mph. Build a model that expresses the distance (d) between thec ars as a funtion of time (t)?
#2 Inscribe a right cylinder of height (h) and radius (r) in a a sphere of fixed radius (R) expressed the volume (V) of the cylinder as a function of (h)
d^2 = (30t)^2 + (40t)^2 = 2500t^2
d = 50 t
(did you notice that the triangle is the 3:4:5 right-angled triangle? )
#2
volume(cylinder) = πr^2 h
make a sketch, let the radius of the cylinder be r
the radius of the sphere R, and the height of the cylinder as h
let 2H = h , that is let the distance from the centre of the sphere to the centre of the cylinder base be h.
then r^2 + H^2 = R^2, where R is a constant
r^2 = R^2 - H^2
Volume = πr^2 H = π(R^2 - H^2)h
= π(R^2H - H^3)
but H = h/2
Vol = π(R^2 h/2 - h^3/8)
or
(1/8)π(4R^2 h - h^3)
check my arithmetic, I should have written it out on paper first.
thanks so much! where di the 3/8 come from??
Sure, I can help you with that!
#1 To build a model that expresses the distance (d) between the cars as a function of time (t), we can use the concept of distance = speed × time.
Let's assume that the two cars start at a point O (intersection) at time t = 0. The car heading south has a constant speed of 30 mph, so its distance from the intersection after time t can be expressed as 30t (since distance = speed × time).
Similarly, the car heading west has a constant speed of 40 mph, so the distance from the intersection after time t can be expressed as 40t.
Now, to find the distance (d) between the two cars, we can use the Pythagorean theorem, as they are moving at right angles to each other (one south and the other west). The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (d) is equal to the sum of the squares of the lengths of the other two sides:
d^2 = (30t)^2 + (40t)^2
Simplifying this equation:
d^2 = 900t^2 + 1600t^2
d^2 = 2500t^2
Taking the square root of both sides:
d = √(2500t^2)
d = 50t
So, the model that expresses the distance (d) between the cars as a function of time (t) is given by d = 50t.
#2 To express the volume (V) of a cylinder as a function of its height (h), knowing that it is inscribed in a sphere of fixed radius (R), we can use the formula for the volume of a cylinder:
V = πr^2h
We also know that the cylinder is inscribed in a sphere, which means that the cylinder's height (h) plus the diameter (D) of the cylinder's base is equal to the diameter of the sphere, which is 2R:
h + D = 2R
Since the diameter is twice the radius, we have:
h + 2r = 2R
Rearranging this equation to express r in terms of R:
r = (2R - h)/2
We can now substitute this value of r into the volume formula:
V = π((2R - h)/2)^2h
Simplifying this equation:
V = π(R - h/2)^2h
Expanding and simplifying:
V = π(R^2 - Rh + h^2/4)h
V = πR^2h - πRh^2 + (πh^3)/4
So, the volume (V) of the cylinder, as a function of its height (h), is given by V = πR^2h - πRh^2 + (πh^3)/4.
I hope that helps! Let me know if you have any further questions.