Henry ran five races, each of which was a different positive integer number of
1 answer:
Answer:
10 miles.
Step-by-step explanation:
Let x be the number of miles on Henry's longest race.
We have been given that Henry ran five races, each of which was a different positive integer number of miles.
We can set an equation for the average of races as:
As distance covered in each race is a different positive integer, so let his first four races be 1, 2, 3, 4.
Now let us substitute the distances of 5 races as:
Let us multiply both sides of our equation by 5.
Let us subtract 10 from both sides of our equation.
Therefore, the maximum possible distance of Henry's longest race is 10 miles.
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Answer:
Step-by-step explanation:
Find the required figure in the attachment. From the figure, <A = <B (alternate interior angle)
Given
∠A=6x+5
∠B=4x+45
Since <A = <B hence;
6x+5 = 4x+45
Collect the like terms
6x-4x = 45-5
2x = 40
Divide both sides by 2;
2x/2 = 40/2
x = 20
Hence the value of x is 20°
Next is to get the measure of <A
Since <A = 6x+5
Substitute x = 20° into the expression
<A = 6(20)+6
<A = 120+6
<A = 126°
Hence the measure of <A is 126°
NOTES:
Degree: the largest exponent in the polynomial
End Behavior:
- Coefficient is POSITIVE, then right side goes to POSITIVE infinity
- Coefficient is NEGATIVE, then right side goes to NEGATIVE infinity
- Degree is EVEN, then left side is SAME direction as right side
- Degree is ODD, then left side is OPPOSITE direction as right side
Multiplicity (M): the exponent of the zero. <em>e.g. (x - 3)² has a multiplicity of 2</em>
Relative max/min: the y-value of the vertices.
- Find the axis of symmetry <em>(the midpoint of two neighboring zeros)</em>
- Plug the x-value from 1 (above) into the given equation to find the y-value. <em>(which is the max/min)</em>
- Repeat 1 and 2 (above) for each pair of neighboring zeros.
Rate of Change: slope between the two given points.
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1. f(x) = (x-1)²(x + 6)
a) Degree = 3
b) end behavior:
- Coefficient is positive so right side goes to positive infinity
- Degree is odd so left side goes to negative infinity
c) (x - 1)²(x + 6) = 0
x - 1 = 0 x + 6 = 0
x = 1 (M=2) x = -6 (M=1)
d) The midpoint between 1 and -6 is -3.5, so axis of symmetry is at x = -3.5
y = (-3.5 - 1)²(-3.5 + 6)
= (-4.5)²(2.5)
= 50.625
50.625 is the relative max
e) see attachment #1
f) The interval at which the graph increases is: (-∞, -3.5)U(1, ∞)
g) The interval at which the graph decreases is: (-3.5, 1)
h) f(-1) = (-1 - 1)²(-1 + 6)
= (-2)²(5)
= 20
f(0) = (0 - 1)²(0 + 6)
= (-1)²(6)
= 6
Find the slope between (-1, 20) and (0, 6)
m =
=
= -14
********************************************************************************************
2. y = x³+3x²-10x
= x(x² + 3x - 10)
= x(x + 5)(x - 2)
a) Degree = 3
b) end behavior:
Coefficient is positive so right side goes to positive infinity
Degree is odd so left side goes to negative infinity
c) x(x + 5)(x - 2) = 0
x = 0 x + 5 = 0 x - 2 = 0
x = 0 (M=1) x = -5 (M=1) x = 2 (M=1)
d) The midpoint between -5 and 0 is -2.5, so axis of symmetry is at x = -2.5
y = -2.5(-2.5 + 5)(-2.5 - 2)
= -2.5(2.5)(-4.5)
= 28.125
28.125 is the relative max
The midpoint between 0 and 2 is 1, so axis of symmetry is at x = 1
y = 1(1 + 5)(1 - 2)
= 1(6)(-1)
= -6
-6 is the relative min
e) see attachment #2
f) The interval at which the graph increases is: (-∞, -2.5)U(1, ∞)
g) The interval at which the graph decreases is: (-2.5, 1)
h) f(-1) = -1(-1 + 5)(-1 - 2)
********************************************************************************************
3. y = -x(x + 2)(x - 7)(x - 3)
a) Degree = 4
b) end behavior:
Coefficient is negative so right side goes to negative infinity
Degree is even so left side goes to negative infinity
c) -x(x + 2)(x - 7)(x - 3) = 0
-x = 0 x + 2 = 0 x - 7 = 0 x - 3 = 0
x = 0 (M=1) x = -2 (M=1) x = 7 (M=1) x = 3 (M=1)
d) The midpoint between -2 and 0 is -1, so axis of symmetry is at x = -1
y = -(-1)(-1 + 2)(-1 - 7)(-1 - 3)
= 1(1)(-8)(-4)
= 32
32 is a relative max
The midpoint between 0 and 3 is 1.5, so axis of symmetry is at x = 1.5
y = -(1.5)(1.5 + 2)(1.5 - 7)(1.5 - 3)
= -1.5(3.5)(-5.5)(-1.5)
= -43.3125
-43.3125 is the relative min
The midpoint between 3 and 7 is 5, so axis of symmetry is at x = 5
y = -(5)(5 + 2)(5 - 7)(5 - 3)
= -5(7)(-2)(2)
= 140
140 is the relative max
e) see attachment #3
f) The interval at which the graph increases is: (-∞, -1)U(1.5, 5)
g) The interval at which the graph decreases is: (-1, 1.5)U(5, ∞)
h) f(-1) = -(-1)(-1 + 2)(-1 - 7)(-1 - 3)
= 1(1)(-8)(-4)
= 32
f(0) = -(0)(0 + 2)(0 - 7)(0 - 3)
= 0
Find the slope between (-1, 32) and (0, 0)
m =
=
= -32
