Math / Algebra

QuestionAssessment Read and analyze the following problems. Shows your complete solutions.
1. In a display window, a grocer wishes to place boxes of detergent in pyramid form so that the bottom row contains 15 boxes, the next row contains 14 boxes. the next row contains 13 boxes, and so on, with one box on top. How many boxes of detergent are necessary for the pyramid?
2. A city has a population of 100,000 . If the population increases 10 percent every 5 years, what will the population be at the end of 40 years?
3. Mr. Vold is a sadistic teacher who likes writing lots of exam questions. He usually starts the semester with only 10 questions on the first exam, but for each subsequent exam he writes one and a half as many questions as were on the previous exam! Since there's no such thing as half a question and Mr. Vold likes writing questions, round your answers up to the next integer. a. How many questions are on the fifth exam of the semester? b. If Mr. Vold wrote 20 exams in a semester, how many total exam questions would they have all together?
4. The table below shows the numbers of days an offshore oil well has been leaking and the diameters (in miles) of the oil spill. Describe the pattern of the numbers of days. \begin{tabular}{|l|c|c|c|c|c|c|} \hline diameter (mi) & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline days & 0 & 6 & 24 & 54 & 96 & 150 \\ \hline \end{tabular} a. Find the model equation of the problem above. b. How long did the oil take to spill if the diameter is 3.5 mi ? 7.5 mi ? c. If the oil took 245 days to spill, what is its diameter?
5. Given the figures below, answer the following questions. \begin{tabular}{|l|c|c|c|c|c|} \hline figure \# & 1 & 2 & 3 & 7 & 18 \\ \hline \# of squares & 5 & 9 & 13 & $?$ & $?$ \\ \hline \end{tabular}
Find the model equation of the problem above. How many squares will there be in the $7^{\mathrm{th}}$ figure? $18^{\text {th }}$ figure? c. In what order of the figure which there will be 61 squares?

Studdy Solution

STEP 1

1. The problems involve arithmetic and geometric sequences, as well as polynomial and linear models.
2. Understanding of basic algebra, geometric series, and arithmetic series is necessary.
3. Some problems may require rounding or approximation.

STEP 2

1. Calculate the total number of boxes in the pyramid.
2. Determine the population of the city after 40 years.
3. Calculate the number of questions on the fifth exam and the total number of questions for 20 exams.
4. Analyze the pattern of the oil spill and find the model equation.
5. Derive the model equation for the number of squares and solve the specific queries.

STEP 3

Identify the arithmetic series representing the number of boxes in each row of the pyramid. The bottom row has 15 boxes, and each row above has one less.
The series is: $15 + 14 + 13 + \ldots + 1$

STEP 4

Use the formula for the sum of the first $n$ terms of an arithmetic series: $S_n = \frac{n}{2} (a + l)$ where $a$ is the first term, $l$ is the last term, and $n$ is the number of terms. Here, $a = 15$ , $l = 1$ , and $n = 15$ .

STEP 5

Calculate the sum: $S_{15} = \frac{15}{2} (15 + 1) = \frac{15}{2} \times 16 = 120$
So, 120 boxes are needed for the pyramid.

STEP 6

Determine the population after 40 years, given a 10% increase every 5 years. Let the initial population $P_0 = 100,000$ .

STEP 7

Use the formula for compound interest: $P = P_0 \left(1 + \frac{r}{100}\right)^n$ where $r = 10$ and $n = \frac{40}{5} = 8$ .

STEP 8

Calculate the population: $P = 100,000 \left(1 + 0.10\right)^8 = 100,000 \times (1.10)^8 = 214,358.88$
Rounding to the nearest whole number, the population will be 214,359.

STEP 9

a Determine the number of questions on the fifth exam. The sequence follows a geometric progression where each term is 1.5 times the previous term.

STEP 10

a Use the formula for the $n$ -th term of a geometric series: $a_n = a \cdot r^{n-1}$ where $a = 10$ , $r = 1.5$ , and $n = 5$ .

STEP 11

a Calculate the fifth term: $a_5 = 10 \times 1.5^4 = 10 \times 5.0625 = 50.625$
Rounding up, there are 51 questions on the fifth exam.

STEP 12

b Determine the total number of questions for 20 exams. Sum the terms of the geometric series.

STEP 13

b Use the sum formula for the first $n$ terms of a geometric series: $S_n = a \frac{1-r^n}{1-r}$

STEP 14

b Calculate the sum for $n = 20$ : $S_{20} = 10 \frac{1-1.5^{20}}{1-1.5} = 10 \frac{1-3,325.256}{-0.5} = 10 \times 6,650.512 = 66,505.12$
Rounding up, the total number of questions is 66,506.

STEP 15

a Analyze the pattern of the oil spill days. The change in days suggests a polynomial relationship.

STEP 16

a Use polynomial regression or inspection to determine the model equation. Notice the pattern: $6 = 1^3 \times 6$ $24 = 2^3 \times 3$ $54 = 3^3 \times 2$ $96 = 4^3 \times 1.5$ $150 = 5^3 \times 1.2$
The pattern suggests: $\text{days} = 6d^2$

STEP 17

b Calculate the days for a 3.5 mi diameter: $\text{days} = 6 \times (3.5)^2 = 6 \times 12.25 = 73.5$

STEP 18

b Calculate the days for a 7.5 mi diameter: $\text{days} = 6 \times (7.5)^2 = 6 \times 56.25 = 337.5$

STEP 19

c Determine the diameter if the oil took 245 days to spill using the inverse of the model equation:
$d = \sqrt{\frac{\text{days}}{6}} = \sqrt{\frac{245}{6}} = \sqrt{40.83} = 6.39$

STEP 20

Identify the model equation for the number of squares. Note the increment pattern which is an arithmetic sequence.

STEP 21

Determine the common difference and use the arithmetic sequence formula. The difference between consecutive terms is 4.

STEP 22

Use the formula for the $n$ -th term of an arithmetic series: $a_n = a + (n-1)d$ where $a = 5$ , $d = 4$ .

STEP 23

Calculate the number of squares in the 7th figure: $a_7 = 5 + (7-1) \times 4 = 5 + 6 \times 4 = 5 + 24 = 29$

STEP 24

Calculate the number of squares in the 18th figure: $a_{18} = 5 + (18-1) \times 4 = 5 + 17 \times 4 = 5 + 68 = 73$

STEP 25

Determine the figure number where there are 61 squares: $61 = 5 + (n-1) \times 4$ Solve for $n$ : $61 - 5 = (n-1) \times 4$ $56 = (n-1) \times 4$ $n - 1 = 14$ $n = 15$

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Studdy Solution

STEP 1

1. The problems involve arithmetic and geometric sequences, as well as polynomial and linear models.2. Understanding of basic algebra, geometric series, and arithmetic series is necessary.3. Some problems may require rounding or approximation.

STEP 2

STEP 3

Identify the arithmetic series representing the number of boxes in each row of the pyramid. The bottom row has 15 boxes, and each row above has one less.The series is: 15+14+13+…+1 15 + 14 + 13 + \ldots + 1 15+14+13+…+1

STEP 4

Use the formula for the sum of the first n n n terms of an arithmetic series: Sn=n2(a+l) S_n = \frac{n}{2} (a + l) Sn​=2n​(a+l) where a a a is the first term, l l l is the last term, and n n n is the number of terms. Here, a=15 a = 15 a=15, l=1 l = 1 l=1, and n=15 n = 15 n=15.

STEP 5

Calculate the sum: S15=152(15+1)=152×16=120 S_{15} = \frac{15}{2} (15 + 1) = \frac{15}{2} \times 16 = 120 S15​=215​(15+1)=215​×16=120So, 120 boxes are needed for the pyramid.

STEP 6

Determine the population after 40 years, given a 10% increase every 5 years. Let the initial population P0=100,000 P_0 = 100,000 P0​=100,000.

STEP 7

Use the formula for compound interest: P=P0(1+r100)n P = P_0 \left(1 + \frac{r}{100}\right)^n P=P0​(1+100r​)n where r=10 r = 10 r=10 and n=405=8 n = \frac{40}{5} = 8 n=540​=8.

STEP 8

Calculate the population: P=100,000(1+0.10)8=100,000×(1.10)8=214,358.88 P = 100,000 \left(1 + 0.10\right)^8 = 100,000 \times (1.10)^8 = 214,358.88 P=100,000(1+0.10)8=100,000×(1.10)8=214,358.88Rounding to the nearest whole number, the population will be 214,359.

STEP 9

a Determine the number of questions on the fifth exam. The sequence follows a geometric progression where each term is 1.5 times the previous term.

STEP 10

a Use the formula for the n n n-th term of a geometric series: an=a⋅rn−1 a_n = a \cdot r^{n-1} an​=a⋅rn−1 where a=10 a = 10 a=10, r=1.5 r = 1.5 r=1.5, and n=5 n = 5 n=5.

STEP 11

a Calculate the fifth term: a5=10×1.54=10×5.0625=50.625 a_5 = 10 \times 1.5^4 = 10 \times 5.0625 = 50.625 a5​=10×1.54=10×5.0625=50.625Rounding up, there are 51 questions on the fifth exam.

STEP 12

b Determine the total number of questions for 20 exams. Sum the terms of the geometric series.

STEP 13

b Use the sum formula for the first n n n terms of a geometric series: Sn=a1−rn1−r S_n = a \frac{1-r^n}{1-r} Sn​=a1−r1−rn​

STEP 14

STEP 15

a Analyze the pattern of the oil spill days. The change in days suggests a polynomial relationship.

STEP 16

STEP 17

b Calculate the days for a 3.5 mi diameter: days=6×(3.5)2=6×12.25=73.5 \text{days} = 6 \times (3.5)^2 = 6 \times 12.25 = 73.5 days=6×(3.5)2=6×12.25=73.5

STEP 18

b Calculate the days for a 7.5 mi diameter: days=6×(7.5)2=6×56.25=337.5 \text{days} = 6 \times (7.5)^2 = 6 \times 56.25 = 337.5 days=6×(7.5)2=6×56.25=337.5

STEP 19

c Determine the diameter if the oil took 245 days to spill using the inverse of the model equation:d=days6=2456=40.83=6.39 d = \sqrt{\frac{\text{days}}{6}} = \sqrt{\frac{245}{6}} = \sqrt{40.83} = 6.39 d=6days​​=6245​​=40.83​=6.39

STEP 20

Identify the model equation for the number of squares. Note the increment pattern which is an arithmetic sequence.

STEP 21

Determine the common difference and use the arithmetic sequence formula. The difference between consecutive terms is 4.

STEP 22

Use the formula for the n n n-th term of an arithmetic series: an=a+(n−1)d a_n = a + (n-1)d an​=a+(n−1)d where a=5 a = 5 a=5, d=4 d = 4 d=4.

STEP 23

Calculate the number of squares in the 7th figure: a7=5+(7−1)×4=5+6×4=5+24=29 a_7 = 5 + (7-1) \times 4 = 5 + 6 \times 4 = 5 + 24 = 29 a7​=5+(7−1)×4=5+6×4=5+24=29

STEP 24

Calculate the number of squares in the 18th figure: a18=5+(18−1)×4=5+17×4=5+68=73 a_{18} = 5 + (18-1) \times 4 = 5 + 17 \times 4 = 5 + 68 = 73 a18​=5+(18−1)×4=5+17×4=5+68=73

STEP 25

Determine the figure number where there are 61 squares: 61=5+(n−1)×4 61 = 5 + (n-1) \times 4 61=5+(n−1)×4 Solve for n n n: 61−5=(n−1)×4 61 - 5 = (n-1) \times 4 61−5=(n−1)×4 56=(n−1)×4 56 = (n-1) \times 4 56=(n−1)×4 n−1=14 n - 1 = 14 n−1=14 n=15 n = 15 n=15

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1. The problems involve arithmetic and geometric sequences, as well as polynomial and linear models.
2. Understanding of basic algebra, geometric series, and arithmetic series is necessary.
3. Some problems may require rounding or approximation.

Identify the arithmetic series representing the number of boxes in each row of the pyramid. The bottom row has 15 boxes, and each row above has one less.
The series is: $15 + 14 + 13 + \ldots + 1$

Use the formula for the sum of the first $n$ terms of an arithmetic series: $S_n = \frac{n}{2} (a + l)$ where $a$ is the first term, $l$ is the last term, and $n$ is the number of terms. Here, $a = 15$ , $l = 1$ , and $n = 15$ .

Calculate the sum: $S_{15} = \frac{15}{2} (15 + 1) = \frac{15}{2} \times 16 = 120$
So, 120 boxes are needed for the pyramid.

Determine the population after 40 years, given a 10% increase every 5 years. Let the initial population $P_0 = 100,000$ .

Use the formula for compound interest: $P = P_0 \left(1 + \frac{r}{100}\right)^n$ where $r = 10$ and $n = \frac{40}{5} = 8$ .

Calculate the population: $P = 100,000 \left(1 + 0.10\right)^8 = 100,000 \times (1.10)^8 = 214,358.88$
Rounding to the nearest whole number, the population will be 214,359.

a Use the formula for the $n$ -th term of a geometric series: $a_n = a \cdot r^{n-1}$ where $a = 10$ , $r = 1.5$ , and $n = 5$ .

a Calculate the fifth term: $a_5 = 10 \times 1.5^4 = 10 \times 5.0625 = 50.625$
Rounding up, there are 51 questions on the fifth exam.

b Use the sum formula for the first $n$ terms of a geometric series: $S_n = a \frac{1-r^n}{1-r}$

b Calculate the days for a 3.5 mi diameter: $\text{days} = 6 \times (3.5)^2 = 6 \times 12.25 = 73.5$

b Calculate the days for a 7.5 mi diameter: $\text{days} = 6 \times (7.5)^2 = 6 \times 56.25 = 337.5$

c Determine the diameter if the oil took 245 days to spill using the inverse of the model equation:
$d = \sqrt{\frac{\text{days}}{6}} = \sqrt{\frac{245}{6}} = \sqrt{40.83} = 6.39$

Use the formula for the $n$ -th term of an arithmetic series: $a_n = a + (n-1)d$ where $a = 5$ , $d = 4$ .

Calculate the number of squares in the 7th figure: $a_7 = 5 + (7-1) \times 4 = 5 + 6 \times 4 = 5 + 24 = 29$

Calculate the number of squares in the 18th figure: $a_{18} = 5 + (18-1) \times 4 = 5 + 17 \times 4 = 5 + 68 = 73$

Determine the figure number where there are 61 squares: $61 = 5 + (n-1) \times 4$ Solve for $n$ : $61 - 5 = (n-1) \times 4$ $56 = (n-1) \times 4$ $n - 1 = 14$ $n = 15$