The formula for the area of a trapezoid is A, equals, one half, h, of, b, start subscript, 1, end subscript, plus, b, start subscript, 2, end subscript, commaA=
2
1
h(b
1
+b
2
), where hh is the height and b, start subscript, 1, end subscriptb
1
and b, start subscript, 2, end subscriptb
2
are the two bases. Rewrite the formula to solve for hh in terms of A, commaA, b, start subscript, 1, end subscriptb
1
and b, start subscript, 2, end subscript, .b
2
.
11 answers
To solve for \( h \) (the height) in terms of \( A \) (the area), \( b_1 \) (the first base), and \( b_2 \) (the second base), we start with the original formula for the area of a trapezoid:
\[
A = \frac{1}{2} h (b_1 + b_2)
\]
To isolate \( h \), we can follow these steps:
Multiply both sides by 2 to eliminate the fraction:
\[
2A = h (b_1 + b_2)
\]
Now, divide both sides by \( (b_1 + b_2) \) to solve for \( h \):
\[
h = \frac{2A}{b_1 + b_2}
\]
So, the formula for the height \( h \) in terms of the area \( A \) and the two bases \( b_1 \) and \( b_2 \) is:
\[
h = \frac{2A}{b_1 + b_2}
\]
The formula for the density of an object is d, equals, start fraction, m, divided by, V, end fraction, commad=
V
m
, where mm is the mass and VV is the volume. Solve the formula for VV in terms of dd and m, .m.
To solve for \( V \) (the volume) in terms of \( d \) (density) and \( m \) (mass), we start with the original formula for density:
\[
d = \frac{m}{V}
\]
To isolate \( V \), we can follow these steps:
Multiply both sides by \( V \) to eliminate the fraction:
\[
dV = m
\]
Now, divide both sides by \( d \) to solve for \( V \):
\[
V = \frac{m}{d}
\]
Thus, the formula for the volume \( V \) in terms of the density \( d \) and the mass \( m \) is:
\[
V = \frac{m}{d}
\]
The formula for the volume of a cylinder is V, equals, pi, r, squared, h, commaV=πr
2
h, where rr is the radius of the cylinder and hh is the height of the cylinder. Rewrite the formula to solve for hh in terms of rr and V, .V.
To solve for \( h \) (the height) in terms of \( r \) (the radius) and \( V \) (the volume) of a cylinder, we start with the original formula for the volume of a cylinder:
\[
V = \pi r^2 h
\]
To isolate \( h \), follow these steps:
Divide both sides by \( \pi r^2 \) to solve for \( h \):
\[
h = \frac{V}{\pi r^2}
\]
Thus, the formula for the height \( h \) in terms of the radius \( r \) and the volume \( V \) is:
\[
h = \frac{V}{\pi r^2}
\]
Identify the property that justifies each step asked about in the answer area below.
Line 1: , q, left parenthesis, p, r, right parenthesis
Line 1: =
q(pr)
Line 2: , left parenthesis, q, p, right parenthesis, r
Line 2: =
(qp)r
Line 3: , left parenthesis, p, q, right parenthesis, r
Line 3: =
(pq)r
Answer
Attempt 1 out of 2
Line 1 to Line 2:
Line 2 to Line 3
To identify the properties that justify the steps from one line to the next in your expressions, we can analyze the equations provided.
Line 1 to Line 2:
\( q(pr) = (qp)r \)
Justification: This step applies the Associative Property of Multiplication. This property states that the way in which factors are grouped in multiplication does not change the product. In this case, we are grouping \( q \) and \( p \) together with \( r \).
Line 2 to Line 3:
\( (qp)r = (pq)r \)
Justification: This step applies the Commutative Property of Multiplication. This property states that the order of factors does not change the product. Here, we are swapping the order of \( q \) and \( p \).
In summary:
Line 1 to Line 2: Associative Property of Multiplication
Line 2 to Line 3: Commutative Property of Multiplication
Identify the property that justifies each step asked about in the answer area below.
Line 1: , left parenthesis, x, plus, 9, right parenthesis, left parenthesis, 4, plus, 3, x, right parenthesis
Line 1: =
(x+9)(4+3x)
Line 2: , left parenthesis, x, plus, 9, right parenthesis, left parenthesis, 3, x, plus, 4, right parenthesis
Line 2: =
(x+9)(3x+4)
Line 3: , left parenthesis, 3, x, plus, 4, right parenthesis, left parenthesis, x, plus, 9, right parenthesis
Line 3: =
(3x+4)(x+9)
Answer
Attempt 1 out of 2
Line 1 to Line 2:
Line 2 to Line 3:
To identify the properties that justify the steps between the lines:
Line 1 to Line 2:
\[
(x + 9)(4 + 3x) = (x + 9)(3x + 4)
\]
Justification: This step uses the Commutative Property of Addition. The order of the terms within the parentheses is changed, but the expression remains equivalent.
Line 2 to Line 3:
\[
(x + 9)(3x + 4) = (3x + 4)(x + 9)
\]
Justification: This step again uses the Commutative Property of Multiplication. The order of the factors is switched, but the product remains the same.
In summary:
Line 1 to Line 2: Commutative Property of Addition
Line 2 to Line 3: Commutative Property of Multiplication
Identify the property that justifies each step asked about in the answer area below.
Line 1: , q, left parenthesis, p, r, right parenthesis
Line 1: =
q(pr)
Line 2: , left parenthesis, q, p, right parenthesis, r
Line 2: =
(qp)r
Line 3: , left parenthesis, p, q, right parenthesis, r
Line 3: =
(pq)r
Answer
Attempt 1 out of 2
Line 1 to Line 2:
Line 2 to Line 3:
To identify the properties that justify the steps between the lines you provided, we analyze each transition:
Line 1 to Line 2:
\( q(pr) = (qp)r \)
Justification: This step applies the Associative Property of Multiplication. The associative property allows us to group factors in a multiplication expression in different ways without changing the product.
Line 2 to Line 3:
\( (qp)r = (pq)r \)
Justification: This step applies the Commutative Property of Multiplication. The commutative property allows us to change the order of the factors in a multiplication expression. Here, \( q \) and \( p \) are swapped.
In summary:
Line 1 to Line 2:Associative Property of Multiplication
Line 2 to Line 3:Commutative Property of Multiplication
1 answer