1. The number of calculators Mrs. Hopkins can buy for the classroom varies inversely as the cost of each calculator. She can buy 24 calculators that cost $60 each. How many calculators can she buy if they cost $80 each?
A: ?
2. Jeffs builds model train layouts. He has $75 to spend on packages of miniature landscape items. He receives 6 free packages with each order. The number of packages y that Jeff can buy is given by y = 75/x + 6, where x represents the cost of each package in dollars. Describe the reasonable domain values and graph the functions.
A: Both the number of packages and the cost of each package in dollars will be nonnegative, so the nonnegative vlaues are reasonable for the domain and range.
Graph: ?
3. Suppose a cone and a cylinder have the same radius and that the slant height l of the cone is the same as the height h of the cylinder. Find the ratio of the cone's surface area to the cylinder's surface area.
Area of cone: S = πrl + πr^2
Area of cylinder: S = 2πrh + 2πr^2
#1 we have n calculators that cost c each.
Since nc = k is constant,
24*60 = 80n
#2
Actually, the cost must be a positive integer divisor of 75, since the number of packages must be an integer.
#3
well, you have the formulas, and you want the ratio
scone/scyl = (πrl + πr^2)/(2πrh + 2πr^2)
= πr(l+r) / 2πr(h+r)
= (l+r) / 2(h+r)
1. To find the number of calculators Mrs. Hopkins can buy if they cost $80 each, we can use the formula for inverse variation:
Number of calculators * Cost per calculator = Constant
Given that Mrs. Hopkins can buy 24 calculators at $60 each, we can set up the following proportion:
24 * 60 = x * 80
Simplifying the equation, we have:
1440 = 80x
Dividing both sides by 80, we find that:
x = 18
So Mrs. Hopkins can buy 18 calculators if they cost $80 each.
2. For the second question, we are given a function y = 75/x + 6, where x represents the cost of each package in dollars and y represents the number of packages Jeff can buy.
To determine the reasonable domain values, we need to consider what values of x make sense in the given context. Since the cost of each package cannot be negative, the domain should be nonnegative (x ≥ 0).
For the range, the number of packages cannot be negative either, so the range is also nonnegative (y ≥ 0).
To graph the function, you can plot points by choosing different values of x and calculating the corresponding values of y based on the equation. This will give you a set of coordinates that can be plotted on a graph. Alternatively, you can use a graphing calculator or software to graph the function directly.
3. To find the ratio of the cone's surface area to the cylinder's surface area, we need to find the expressions for the surface areas of both objects.
The surface area of a cone is given by:
Surface area of cone (S) = πrl + πr^2
The surface area of a cylinder is given by:
Surface area of cylinder (S) = 2πrh + 2πr^2
Since the radius (r) and slant height (l) of the cone are the same as the radius (r) and height (h) of the cylinder, we can substitute those values into the formulas:
Surface area of cone (S) = πrl + πr^2 = πrh + πr^2
Surface area of cylinder (S) = 2πrh + 2πr^2
To find the ratio of the cone's surface area to the cylinder's surface area, divide the surface area of the cone by the surface area of the cylinder:
Ratio = (πrh + πr^2) / (2πrh + 2πr^2)
Simplifying the equation, we find:
Ratio = (πrh + πr^2) / (2πrh + 2πr^2) = (rh + r^2) / (2rh + 2r^2) = (h + r) / (2h + 2r)
So, the ratio of the cone's surface area to the cylinder's surface area is (h + r) / (2h + 2r).