[T] The concentration of hydrogen ions in a substance is denoted by [ H + ] , measured in moles per liter. The pH of a substance is defined by the logarithmic function p H = − log [ H + ] . This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base. Find the pH of the following substances. Round answers to one digit. Determine whether the substance is an acid or a base. i . E g g s : [ H + ] = 1.6 × 10 − 8 m o l / L i i . B e e r : [ H + ] = 3.16 × 10 − 3 m o l / L i i i . T o m a t o J u i c e : [ H + ] = 7.94 × 10 − 5 m o l / L
[T] The concentration of hydrogen ions in a substance is denoted by [ H + ] , measured in moles per liter. The pH of a substance is defined by the logarithmic function p H = − log [ H + ] . This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base. Find the pH of the following substances. Round answers to one digit. Determine whether the substance is an acid or a base. i . E g g s : [ H + ] = 1.6 × 10 − 8 m o l / L i i . B e e r : [ H + ] = 3.16 × 10 − 3 m o l / L i i i . T o m a t o J u i c e : [ H + ] = 7.94 × 10 − 5 m o l / L
[T] The concentration of hydrogen ions in a substance is denoted by
[
H
+
]
, measured in moles per liter. The pH of a substance is defined by the logarithmic function
p
H
=
−
log
[
H
+
]
. This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.
Find the pH of the following substances. Round answers to one digit.
Determine whether the substance is an acid or a base.
i
.
E
g
g
s
:
[
H
+
]
=
1.6
×
10
−
8
m
o
l
/
L
i
i
.
B
e
e
r
:
[
H
+
]
=
3.16
×
10
−
3
m
o
l
/
L
i
i
i
.
T
o
m
a
t
o
J
u
i
c
e
:
[
H
+
]
=
7.94
×
10
−
5
m
o
l
/
L
A ladder 27 feet long leans against a wall and the foot of the ladder is sliding away at a constant rate of 3 feet/sec. Meanwhile, a firefighter is climbing up the ladder at a rate of 2 feet/sec. When the firefighter has climbed up 6 feet of the ladder, the ladder makes an angle of л/3 with the ground. Answer the two related
rates questions below. (Hint: Use two carefully labeled similar right triangles.)
(a) If h is the height of the firefighter above the ground, at the instant the angle of the ladder with the ground is л/3, find dh/dt=
feet/sec.
(b) If w is the horizontal distance from the firefighter to the wall, at the instant the angle of the ladder with the ground is л/3, find dw/dt=
feet/sec.
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate (in mi/h) is the distance between the cars increasing four hours later?
Step 1
Using the diagram of a right triangle given below, the relation between x, y, and z is
z²
= x²+
+12
x
Step 2
We must find dz/dt. Differentiating both sides and simplifying gives us the following.
2z
dz
dt
dx
2x.
+2y
dt
dx
dy
dz
x
+y
dt
dt
dt
2z
dy
dt
×
dx
(x+y
dt
dy
dt
An elastic rope is attached to the ground at the positions shown in the picture. The rope is being pulled up along the dotted line. Assume the units are meters.
9
ground level
Assume that x is increasing at a rate of 3 meters/sec.
(a) Write as a function of x: 0=
(b) When x=10, the angle is changing at a rate of
rad/sec.
(c) Let L be the the left hand piece of rope and R the right hand piece of rope. When x=10, is the rate of change of L larger than the rate of change of R?
○ Yes
○ No
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