8th Grade Math Summer (8thGradeMathSummer)

Transcription

1 Name: Date: 1. The cube shown has an edge length of x inches. The equation can be used to determine the length, in inches, of each edge of the cube. What is the value of x? Explain your answer or show your work. Convert the value of x from inches to feet. Write your answer in the form of Show your work. where y and n are both integers. The expression shown can be used to determine the volume of the cube when the length of each edge of the cube is tripled. Let a represent the length of each edge of the cube. Determine the value of b in the expression. Part D Using your answer from or to represent a and the value you found for b, rewrite the expression Part E Evaluate the expression you wrote in Part D. Label your answer with the correct units. 1

2 2. The square shown in the diagram has a side of length s units. The area is at least 15 square units and at most 50 square units. What exact interval contains all possible values of s? Show your work or explain your answer. On a number line, plot points that represent the greatest and least values of s. Be as precise as possible. What is the approximate unit distance between the two points that you plotted? Explain your answer. 3. Which of the following best represents? A number between A. 3 and 4 B. 6 and 7 C. 7 and 8 D. 8 and 10 Permission has been granted for reproduction by the Virginia Department of Education Virginia Department of Education 2

3 4. A warehouse stores goods in cube-shaped boxes, each with a volume of cubic feet. If the volume of a single box is 216 cubic centimeters, what is the value of x? Explain your answer. In one room, the boxes are arranged together to form a rectangular solid measuring feet high, feet long, and feet wide. How many boxes are arranged together in this room? Explain your answer. In a second room, boxes are arranged together in a straight line. What is the total volume of all the boxes in the second room in terms of x? Explain your answer. 3

4 5. The word micron is an abbreviated term for micrometer, or of one meter. Write an expression in scientific notation that is equivalent to Show your work. Find the area, in square microns, of a rectangular object with a width of 6,000 microns and a length of 2,000,000 microns. Show your work and write your final answer in scientific notation. The area of a different figure can be expressed as square microns. What is this area in square meters? Show your work and write your final answer in scientific notation. Part D The length of a virus is 0.4 microns. How many times the length of the virus is a 40-meter plastic tube? Show your work. 6. The diameter of an atom is centimeters. The diameter of the nucleus of the atom is centimeters. How many times longer is the diameter of the atom than the diameter of its nucleus? Show your work. To convert a length from centimeters to nanometers, the quantity is multiplied by nucleus, in nanometers? Show your work. What is the diameter of the 4

5 7. The table shows the population estimates for 3 countries in scientific notation. What number, written in scientific notation, represents the combined total population of countries P and Q? Show your work or explain your answer. How many times greater is the population of Q than the population of R? Show your work or explain your answer. The population of P is predicted to increase by 10% during the next 20 years. What number, written in scientific notation, represents the predicted total population of P? Show your work. Part D The population of R is twice the population it was 40 years ago. What was the average gain per year in population during this time period? Show your work. 8. Which expression below represents in scientific notation? A B C D

6 9. The diameter of Earth is approximately 13,000 kilometers. The diameter of a grain of sand is approximately millimeter. 8th Grade Math Summer (8thGradeMathSummer) If the grain of sand is spherical, what would its volume be in cubic millimeters? Use 3 to approximate your answer in scientific notation. and write The number of millimeters equal to 13,000 kilometers is How many times greater is the diameter of Earth than the diameter of the grain of sand? Show your work, and write your answer in scientific notation. The diameter of a volleyball is approximately 21.5 centimeters, or kilometers. To the nearest whole number, how many volleyballs, placed side by side, would be needed to equal the length of the diameter of Earth? Show your work. 6

7 10. A biology teacher gave three students in his class a different plant. He asked the students to record the heights of their plants, in centimeters (cm), everyday for one week. The students reported their results in different displays. Their results are shown in a graph, an equation, and a table. List the plants in order from the slowest rate of growth per day to the greatest rate of growth per day. Explain how you got your answer. If the growth rate for Plant 1 remains the same, on what day number will it be 24 cm tall? Explain how you got your answer. If the growth rate for each plant remains the same, when will Plant 2 and Plant 3 be the same height? Show work or give an explanation to justify your answer. 7

8 11. The graph shows the total number of miles, y, Miranda traveled in x hours. The total number of miles, y, Claude traveled in x hours is given by the equation comparing his speed and Miranda s speed is true? Which statement A. Miranda s speed is greater since B. Miranda s speed is greater since C. Claude s speed is greater since D. Claude s speed is greater since 8

9 12. Lines m and n are graphed on a coordinate grid. 8th Grade Math Summer (8thGradeMathSummer) Line m has a slope of 2 and a y -intercept of Line n has a slope of 3 and a y -intercept of 2. Write the equation that represents line m. Write the equation that represents line n. Using the equations you wrote in, find the point where lines m and n intersect. Show your work. Write the equation of a line that passes through both the intersection point found in and the origin. Explain your equation or show your work. 9

10 13. The line shown on the coordinate plane contains points X, Y, and Z. Locate two points, V and W, such that and are overlapping right triangles, with as one hypotenuse and as the other. What are the coordinates for V and W? Show that the two overlapping right triangles from are similar triangles by comparing the ratios of each triangle s vertical leg length to its horizontal leg length. What is true about these two ratios? Explain your answer. Use the coordinates of the endpoints of and to determine the slope of each hypotenuse. Show your work. Part D How do your answers to and relate? Explain your answer. 10

11 14. If what is the value of x? A. B. C. 1 D A wedding cake has three layers. The bottom layer is x inches in height. Each layer above is 1 inch greater than the height of the layer below it. The height of each layer is shown in the diagram. If the total height of the cake is inches, how tall is the bottom layer? A. B. C. D. 11

12 16. Given the system of the following linear equations: 8th Grade Math Summer (8thGradeMathSummer) 2x 3y = 12 5x + 4y = 7 Solve for x and y. A. (-2, 3) B. (-2, -3) C. (3, 2) D. (3, -2) 17. Jean surveyed 25% of all the eighth graders and 75% of all the seventh graders at her school. Keisha surveyed 50% of all the eighth graders and 50% of all the seventh graders at the same school. Jean surveyed a total of 152 students. Keisha surveyed a total of 144 students. What is the number of eighth graders at Jean and Keisha s school? A. 110 B. 128 C. 160 D

13 8th Grade Math Summer (8thGradeMathSummer) 18. Liam used the equation shown to create a sequence where represents the first term in the sequence. Create a table to represent the first 5 terms of the sequence. Liam says the equation can be used to represent a functional relationship. Explain whether or not Liam s statement is true. Martha attempts to graph the equation Liam used to create a sequence. Her graph of the equation is shown below. Martha says the equation Liam used for the sequence represents a linear relationship. Martha is incorrect. Write the equation that represents Martha s graph. Explain whether or not Martha s graph represents a linear relationship. Explain the error Martha made when creating the graph of the equation Liam used to create a sequence. 13

14 19. The equation represents the depth, in feet, of a submarine diving below the ocean s surface, where d represents the depth and t represents the time, in minutes. A man lowers a lobster trap below the ocean s surface at a constant rate of 30 feet per minute. How does the rate of change of the lobster trap compare to the rate of change of the submarine? Explain your answer. At what time does the lobster trap begin its descent? How many feet is the lobster trap below the ocean s surface at this time? Write an equation representing the depth, in feet, of the lobster trap in terms of the time. Part D Will the graphs of these two functions intersect? Explain your answer. 14

15 20. Bruce pays rent for a storage unit each month. The company charges Bruce a fixed rate per month with an initial fee for moving the items into the storage unit. The table shows the total amount in dollars, t, Bruce has paid to the company after m months. Rita also rents a storage unit, but from a different company. The total amount in dollars, t, Rita pays after m months is represented by the equation Identify two differences between the rental plans represented by the table and the equation. Explain your answers. Using the data in the table, write an equation that can be used to find t, the total amount Bruce pays for m months. Explain what each term in the equation represents. If both Bruce and Rita continue to rent their storage units at the same rates, which of them will pay a total of $1,200 first? Show your work. 15

16 21. A line is shown on the coordinate grid below. Which of the following best represents an equation of this line? A. B. C. D. This online assessment item contains material that has been released to the public by the Massachusetts Department of Education. 16

17 22. A given function is defined by the equation 8th Grade Math Summer (8thGradeMathSummer) Does the equation define a linear function? Explain your answer. Two functions are considered parallel if their slopes are equivalent. Write the equation for a function that is parallel to the given function with a y -intercept of you wrote. Explain the equation Write the equation for a function that represents a proportional relationship and is also parallel to the given function. Explain the equation you wrote. 23. The function estimates the area, A square inches, of a circle with radius r inches. Use the formula to determine the value of A for and Show your work. Draw a first-quadrant coordinate grid, where the horizontal axis represents the radius and the vertical axis represents the area of the circle. Plot the points for the function that you found in. Is this function linear or nonlinear? Explain your answer. 17

18 24. The coordinate grid shows the reading rates of Justin and Nadia during a reading program at school. Based on the graph, how long will it take Nadia to read 1 book? Explain your answer. If Justin s reading rate stays the same, how many books will Justin read in two weeks? Explain your answer. Based on the graph, what does the y -intercept for Nadia s reading rate represent? Using the graph, compare the y -intercept for Justin s reading rate to the y -intercept for Nadia s reading rate. Explain your answer. Part D Rafael is also in the reading program with Nadia and Justin. The table shows Rafael s reading rate over a period of six weeks. 18

19 Write a function to represent Rafael s reading rate as shown in the table. Show your work. Who has the greatest reading rate (Justin, Nadia, or Rafael)? Explain your answer. 25. Which is the function described by the table of ordered pairs? A. y = x + 1 B. y = 3x C. y = 2x + 3 D. y = x + 13 Permission has been granted for reproduction by the Virginia Department of Education Virginia Department of Education 19

20 Answer Key B) 6 and A) D) Claude s speed is greater since A) 15. B) 16. D) (3, -2) 17. B) C) 20

21 C) y = 2x