int math 2 section 6-8 1011
DESCRIPTION
Direct VariationTRANSCRIPT
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Section 6-8Direct Variation
Wednesday, April 11, 2012
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Essential Question
✦ How do you solve problems involving direct variation and direct square variation functions?
✦ Where you’ll see this:
✦ Finance, biology, physics, business
Wednesday, April 11, 2012
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Vocabulary1. Direct Variation:
2. Constant of Variation:
3. Direct Square Variation:
Wednesday, April 11, 2012
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Vocabulary1. Direct Variation: A function where when one value
gets larger, so does the other; occurs where only multiplication is being applied to the independent variable
2. Constant of Variation:
3. Direct Square Variation:
Wednesday, April 11, 2012
![Page 5: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/5.jpg)
Vocabulary1. Direct Variation: A function where when one value
gets larger, so does the other; occurs where only multiplication is being applied to the independent variable
2. Constant of Variation: A ratio that is found in a direct variation problem by dividing a range value (y) by its corresponding domain value (x)
3. Direct Square Variation:
Wednesday, April 11, 2012
![Page 6: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/6.jpg)
Vocabulary1. Direct Variation: A function where when one value
gets larger, so does the other; occurs where only multiplication is being applied to the independent variable
2. Constant of Variation: A ratio that is found in a direct variation problem by dividing a range value (y) by its corresponding domain value (x)
3. Direct Square Variation: A direct variation problem that produces a parabola due to squaring the independent variable
Wednesday, April 11, 2012
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Direct Variation
Wednesday, April 11, 2012
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Direct Variation
y = kx
Wednesday, April 11, 2012
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Direct Variation
y = kx
k ≠ 0
Wednesday, April 11, 2012
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Direct Variation
y = kx
k ≠ 0
(0, 0)
Wednesday, April 11, 2012
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Direct Variation
y = kx
k ≠ 0
(0, 0)
Line
Wednesday, April 11, 2012
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Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Wednesday, April 11, 2012
![Page 13: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/13.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Wednesday, April 11, 2012
![Page 14: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/14.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation:
Wednesday, April 11, 2012
![Page 15: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/15.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kx
Wednesday, April 11, 2012
![Page 16: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/16.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperature
Wednesday, April 11, 2012
![Page 17: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/17.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
Wednesday, April 11, 2012
![Page 18: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/18.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
y = kx
Wednesday, April 11, 2012
![Page 19: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/19.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
y = kx
559 = k(40)
Wednesday, April 11, 2012
![Page 20: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/20.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
y = kx
559 = k(40)40 40
Wednesday, April 11, 2012
![Page 21: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/21.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
y = kx
559 = k(40)40 40k = 13.975
Wednesday, April 11, 2012
![Page 22: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/22.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
y = kx
559 = k(40)40 40k = 13.975
y = 13.975x
Wednesday, April 11, 2012
![Page 23: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/23.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
y = kx
559 = k(40)40 40k = 13.975
y = 13.975x
y = 13.975(25)
Wednesday, April 11, 2012
![Page 24: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/24.jpg)
Example 1A law discovered by Jacques Charles, a French scientist of the late 18th and early 19th centuries, states that for a
fixed amount of gas, the volume varies directly as the temperature. If a gas has a volume of 559 mL at 40°C,
what is the volume at 25°C?
Direct variation: y = kxvolume temperaturey = x =
y = kx
559 = k(40)40 40k = 13.975
y = 13.975x
y = 13.975(25)y = 349.375 mL
Wednesday, April 11, 2012
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Direct Square Variation
Wednesday, April 11, 2012
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Direct Square Variation
y = kx2
Wednesday, April 11, 2012
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Direct Square Variation
y = kx2
k ≠ 0
Wednesday, April 11, 2012
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Direct Square Variation
y = kx2
k ≠ 0
(0, 0)
Wednesday, April 11, 2012
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Direct Square Variation
y = kx2
k ≠ 0
(0, 0)
Parabola
Wednesday, April 11, 2012
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Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Wednesday, April 11, 2012
![Page 31: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/31.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Wednesday, April 11, 2012
![Page 32: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/32.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation:
Wednesday, April 11, 2012
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Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2
Wednesday, April 11, 2012
![Page 34: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/34.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speed
Wednesday, April 11, 2012
![Page 35: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/35.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
Wednesday, April 11, 2012
![Page 36: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/36.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =27 m = .027 km
Wednesday, April 11, 2012
![Page 37: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/37.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
Wednesday, April 11, 2012
![Page 38: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/38.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
Wednesday, April 11, 2012
![Page 39: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/39.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)
Wednesday, April 11, 2012
![Page 40: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/40.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)4096 4096
Wednesday, April 11, 2012
![Page 41: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/41.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)4096 4096
k = .000006591796875
Wednesday, April 11, 2012
![Page 42: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/42.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)4096 4096
k = .000006591796875
k ≈ 6.59 ×10−6
Wednesday, April 11, 2012
![Page 43: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/43.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)4096 4096
k = .000006591796875
k ≈ 6.59 ×10−6
y ≈ 6.59 ×10−6 x2
Wednesday, April 11, 2012
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Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)4096 4096
k = .000006591796875
k ≈ 6.59 ×10−6
y ≈ 6.59 ×10−6 x2
y ≈ 6.59 ×10−6(90)2
Wednesday, April 11, 2012
![Page 45: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/45.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)4096 4096
k = .000006591796875
k ≈ 6.59 ×10−6
y ≈ 6.59 ×10−6 x2
y ≈ 6.59 ×10−6(90)2
y ≈ .053379 km
Wednesday, April 11, 2012
![Page 46: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/46.jpg)
Example 2The distance it takes to stop after you have applied the
brakes of a car varies directly as the square of the speed of the car. A car going 64 km/h stops in 27 m. If you are
going 90 km/h, how far will the car travel after you have applied the brakes?
Direct variation: y = kx2 distance speedy = x =
y = kx227 m = .027 km
.027 = k(64)2
.027 = k(4096)4096 4096
k = .000006591796875
k ≈ 6.59 ×10−6
y ≈ 6.59 ×10−6 x2
y ≈ 6.59 ×10−6(90)2
y ≈ .053379 km
y ≈ 53.379 m
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
a. When x = 10, y = 15. Find y when x = 15.
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
a. When x = 10, y = 15. Find y when x = 15.
y = kx
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
a. When x = 10, y = 15. Find y when x = 15.
y = kx
15 = k(10)
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
a. When x = 10, y = 15. Find y when x = 15.
y = kx
15 = k(10)
k =
32
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
a. When x = 10, y = 15. Find y when x = 15.
y = kx
15 = k(10)
k =
32
y =
32
x
Wednesday, April 11, 2012
![Page 52: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/52.jpg)
Example 3Assume y varies directly as x.
a. When x = 10, y = 15. Find y when x = 15.
y = kx
15 = k(10)
k =
32
y =
32
x
y =
32
(15)
Wednesday, April 11, 2012
![Page 53: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/53.jpg)
Example 3Assume y varies directly as x.
a. When x = 10, y = 15. Find y when x = 15.
y = kx
15 = k(10)
k =
32
y =
32
x
y =
32
(15)
y =
452
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
b. When x = 3, y = 12. Find y when x = 21.
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
b. When x = 3, y = 12. Find y when x = 21.
y = kx
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
b. When x = 3, y = 12. Find y when x = 21.
y = kx
12 = k(3)
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
b. When x = 3, y = 12. Find y when x = 21.
y = kx
12 = k(3)
k = 4
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
b. When x = 3, y = 12. Find y when x = 21.
y = kx
12 = k(3)
k = 4
y = 4x
Wednesday, April 11, 2012
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Example 3Assume y varies directly as x.
b. When x = 3, y = 12. Find y when x = 21.
y = kx
12 = k(3)
k = 4
y = 4x
y = 4(21)
Wednesday, April 11, 2012
![Page 60: Int Math 2 Section 6-8 1011](https://reader037.vdocument.in/reader037/viewer/2022102902/559b54db1a28ab810c8b4630/html5/thumbnails/60.jpg)
Example 3Assume y varies directly as x.
b. When x = 3, y = 12. Find y when x = 21.
y = kx
12 = k(3)
k = 4
y = 4x
y = 4(21)
y = 84
Wednesday, April 11, 2012
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Problem Set
Wednesday, April 11, 2012
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Problem Set
p. 278 #1-23 odd
"The conventional view serves to protect us from the painful job of thinking." – John Kenneth Galbraith
Wednesday, April 11, 2012