Question

the expression $200 + 4(d - 3)^2$ models the cost of a service after $d$ days. the constant in the simplified form of the expression represents the initial cost, in dollars. what is the initial cost of the service?
(a) $164
(b) $200
(c) $216
(d) $236
6.
a car rental agency charges a fixed daily fee plus a cost per mile driven. the expression $45 + 0.50m$ represents the total daily cost in dollars, where $m$ is the number of miles driven. which statements are true?

1. select two answer choices.

a. the value 0.50 represents the fixed daily fee.
b. the value 0.50 represents the cost per mile.
c. the value 45 represents the cost per mile.
d. the value 45 represents the fixed daily fee.
e. the value $m$ represents the total cost.
8.
a large swimming pool is being filled with water. the expression $2000 + 400h$ represents the volume of water in gallons after $h$ hours. which statements are true?
select two answer choices.
a. the value 400 represents the rate at which the pool is filled per hour in gallons.
b. the value 400 represents the initial volume of water in gallons.
c. the value 2000 represents the rate at which the pool is filled per hour in gallons.
d. the value 2000 represents the initial volume of water in the pool in gallons.
objective:a-sse.2 use the structure of an expression to identify ways to rewrite it. for example, see $x^4 - y^4$ as $(x^2)^2 - (y^2)^2$, thus recognizing it as a difference of squares that can be factored as

Explanation:

Question 5 (a - d)

Step1: Identify initial cost condition

Initial cost occurs at \( d = 0 \) (day 0, start). Substitute \( d = 0 \) into \( 200 + 4(d - 3)^2 \).

Step2: Substitute and calculate

Substitute \( d = 0 \): \( 200 + 4(0 - 3)^2 = 200 + 4\times(-3)^2 = 200 + 4\times9 = 200 + 36 = 236 \)? Wait, no—wait, the problem says "the constant in the simplified form". Wait, maybe I misread. Wait, the expression is \( 200 + 4(d - 3)^2 \). To simplify, expand \( (d - 3)^2 = d^2 - 6d + 9 \), then \( 4(d^2 - 6d + 9) = 4d^2 - 24d + 36 \), then total expression is \( 4d^2 - 24d + 36 + 200 = 4d^2 - 24d + 236 \). Wait, but initial cost is when \( d = 0 \), so plug \( d = 0 \) into original: \( 200 + 4(0 - 3)^2 = 200 + 49 = 200 + 36 = 236 \)? But option (d) is 236? Wait, no, maybe the problem is "the constant in the simplified form"—wait, maybe I made a mistake. Wait, no, let's re - check. Wait, the problem says "the constant in the simplified form of the expression represents the initial cost". Wait, when we simplify \( 200 + 4(d - 3)^2 \), first expand \( (d - 3)^2 = d^2 - 6d + 9 \), multiply by 4: \( 4d^2 - 24d + 36 \), then add 200: \( 4d^2 - 24d + 236 \). So the constant term is 236? But option (d) is 236. Wait, but maybe the problem is "initial cost" which is when \( d = 3 \)? No, initial cost is at \( d = 0 \). Wait, maybe the problem has a typo, or I misread. Wait, no, let's check again. The expression is \( 200 + 4(d - 3)^2 \). If \( d = 3 \), then \( (d - 3)^2 = 0 \), so cost is 200. Wait, maybe the problem means "the constant term when \( d = 3 \)"? No, initial cost is at the start, \( d = 0 \). Wait, there is a contradiction here. Wait, maybe the problem is "the constant in the simplified form"—wait, when \( d = 0 \), the value is \( 200 + 4(9)=236 \), but if we consider the simplified expression \( 4d^2 - 24d + 236 \), the constant term is 236. But let's check the options: (a) 164, (b) 200, (c) 216, (d) 236. So the answer should be (d) $236? Wait, but maybe I made a mistake in expansion. Wait, \( (d - 3)^2 = d^2 - 6d + 9 \), \( 4(d^2 - 6d + 9)=4d^2 - 24d + 36 \), \( 200 + 4d^2 - 24d + 36 = 4d^2 - 24d + 236 \). So the constant term is 236, so initial cost is 236, which is option (d).

The expression for total daily cost is \( 45+0.50m \), where \( m \) is the number of miles driven. In a linear cost model of the form \( \text{Total Cost}=\text{Fixed Cost}+\text{Variable Cost per Unit}\times\text{Number of Units} \):

• The fixed daily fee is the constant term (the term that does not depend on the number of miles), so 45 is the fixed daily fee (so option D is correct).
• The cost per mile is the coefficient of \( m \) (the variable representing the number of miles), so 0.50 is the cost per mile (so option B is correct).
• Option A is wrong because 0.50 is not the fixed fee. Option C is wrong because 45 is not the cost per mile. Option E is wrong because \( m \) is the number of miles, not the total cost.

The expression for the volume of water is \( 2000 + 400h \), where \( h \) is the number of hours. In a linear model of the form \( \text{Total Volume}=\text{Initial Volume}+\text{Rate of Filling per Hour}\times\text{Number of Hours} \):

• The rate at which the pool is filled per hour is the coefficient of \( h \), so 400 is the rate (so option A is correct).
• The initial volume of water (when \( h = 0 \), at the start) is the constant term, so 2000 is the initial volume (so option D is correct).
• Option B is wrong because 400 is not the initial volume. Option C is wrong because 2000 is not the rate.

Answer:

(d) $236

Question 7 (A - E)