#1 The cost of butter is fixed. If 5lb. cost $4.00, how much will 7 1/2 lb cost?
#2 Y varies directly as X; That is Y/X = M. If Y = 6 when X = 3,
find Y when X = 11.
#3 In any triangle, A = 1/2 bh. When b is constant,
(a) Does A vary directly as h? Why?
(b) If h is halved, what happens to A?
(c) If A = 30 when h = 10, find A when h = 20.
d) If A = 40 when h = 6 2/3, find h when A = 50.
#5 If the lengths of the bases of a trapezoid are constant, does the area
vary directly as the length of the altitude? Why?
#1 To find how much 7 1/2 lb of butter will cost, you can use a proportion.
Since 5 lb of butter costs $4.00, you can set up the proportion:
5 lb / $4.00 = 7.5 lb / x
To solve for x (the cost of 7 1/2 lb of butter), cross multiply:
5 lb * x = $4.00 * 7.5 lb
Simplifying:
5x = $30.00
Dividing both sides by 5:
x = $6.00
Therefore, 7 1/2 lb of butter will cost $6.00.
#2 Y varies directly as X, which can be expressed as the equation Y = MX, where M is the constant of variation.
Given that Y = 6 when X = 3, you can substitute these values into the equation and solve for M:
6 = M * 3
Dividing both sides by 3:
M = 6 / 3
M = 2
Now that you know M, you can find Y when X = 11:
Y = 2 * 11
Y = 22
Therefore, when X = 11, Y will be equal to 22.
#3 In any given triangle, the area (A) is equal to one-half the product of the base (b) and the height (h), written as A = 1/2bh.
(a) Yes, the area (A) varies directly with the height (h). This is because the area of a triangle is directly proportional to the height and the base, as given by the formula A = 1/2bh.
(b) If the height (h) is halved, the area (A) will also be halved. This is because the formula for the area of a triangle is dependent on the product of the height (h) and the base (b). If either of these values is halved, the overall area will also be halved.
(c) If A = 30 when h = 10, you can substitute these values into the formula and solve for b:
30 = 1/2 * b * 10
Multiplying both sides by 2:
60 = 10b
Dividing both sides by 10:
6 = b
Therefore, when h = 20, the base (b) will be equal to 6, and the area (A) can be calculated using the formula A = 1/2bh.
A = 1/2 * 6 * 20
A = 60
Therefore, when h = 20, the area (A) will be equal to 60.
(d) If A = 40 when h = 6 2/3, you can substitute these values into the formula and solve for b:
40 = 1/2 * b * (20/3)
Multiplying both sides by 2:
80 = b * (20/3)
Dividing both sides by (20/3), or multiplying by its reciprocal (3/20):
80 * (3/20) = b
12 = b
Therefore, when A = 50, the base (b) will be equal to 12, and the height (h) can be calculated using the formula A = 1/2bh.
50 = 1/2 * 12 * h
Dividing both sides by 6:
8.33 = h
Therefore, when A = 50, the height (h) will be approximately 8.33.
#5 No, the area of a trapezoid does not vary directly with the length of the altitude. The area of a trapezoid is calculated using the formula A = 1/2h(a + b), where h is the length of the altitude and a and b are the lengths of the bases.
In this formula, the area is dependant on the sum of the lengths of the bases (a + b), not solely on the length of the altitude (h). Therefore, the area can vary even if the length of the altitude remains constant.